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mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Computer science</span> </div> </a> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linguistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linguistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Linguistics</span> </div> </a> <ul id="toc-Linguistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physics_and_chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics_and_chemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Physics and chemistry</span> </div> </a> <ul id="toc-Physics_and_chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Social_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Social_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Social sciences</span> </div> </a> <ul id="toc-Social_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Biology</span> </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Mathematics</span> </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_topics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Other topics</span> </div> </a> <ul id="toc-Other_topics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Representation</span> </div> </a> <button aria-controls="toc-Representation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Representation subsection</span> </button> <ul id="toc-Representation-sublist" class="vector-toc-list"> <li id="toc-Visual:_Graph_drawing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visual:_Graph_drawing"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Visual: Graph drawing</span> </div> </a> <ul id="toc-Visual:_Graph_drawing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tabular:_Graph_data_structures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tabular:_Graph_data_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Tabular: Graph data structures</span> </div> </a> <ul id="toc-Tabular:_Graph_data_structures-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Problems</span> </div> </a> <button aria-controls="toc-Problems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Problems subsection</span> </button> <ul id="toc-Problems-sublist" class="vector-toc-list"> <li id="toc-Enumeration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Enumeration"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Enumeration</span> </div> </a> <ul id="toc-Enumeration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgraphs,_induced_subgraphs,_and_minors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subgraphs,_induced_subgraphs,_and_minors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Subgraphs, induced subgraphs, and minors</span> </div> </a> <ul id="toc-Subgraphs,_induced_subgraphs,_and_minors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_coloring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_coloring"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Graph coloring</span> </div> </a> <ul id="toc-Graph_coloring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subsumption_and_unification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subsumption_and_unification"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Subsumption and unification</span> </div> </a> <ul id="toc-Subsumption_and_unification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Route_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Route_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Route problems</span> </div> </a> <ul id="toc-Route_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Network_flow" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Network_flow"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Network flow</span> </div> </a> <ul id="toc-Network_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Visibility_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visibility_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Visibility problems</span> </div> </a> <ul id="toc-Visibility_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covering_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Covering_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Covering problems</span> </div> </a> <ul id="toc-Covering_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decomposition_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decomposition_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Decomposition problems</span> </div> </a> <ul id="toc-Decomposition_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_classes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.10</span> <span>Graph classes</span> </div> </a> <ul id="toc-Graph_classes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Subareas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subareas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Subareas</span> </div> </a> <ul id="toc-Subareas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Online_textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online_textbooks"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Online textbooks</span> </div> </a> <ul id="toc-Online_textbooks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Graph theory</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 72 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-72" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">72 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%8A%90_%E1%8C%8D%E1%88%AB%E1%8D%8D" title="ሥነ ግራፍ – Amharic" lang="am" hreflang="am" data-title="ሥነ ግራፍ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%A8%D9%8A%D8%A7%D9%86" title="نظرية البيان – Arabic" lang="ar" hreflang="ar" data-title="نظرية البيان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos – Aragonese" lang="an" hreflang="an" data-title="Teoría de grafos" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos – Asturian" lang="ast" hreflang="ast" data-title="Teoría de grafos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qraf_n%C9%99z%C9%99riyy%C9%99si" title="Qraf nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Qraf nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%AB_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="গ্রাফ তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="গ্রাফ তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D1%84%D1%82%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Графтар теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Графтар теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B3%D1%80%D0%B0%D1%84%D0%B0%D1%9E" title="Тэорыя графаў – Belarusian" lang="be" hreflang="be" data-title="Тэорыя графаў" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D1%84%D0%B8%D1%82%D0%B5" title="Теория на графите – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на графите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorija_grafova" title="Teorija grafova – Bosnian" lang="bs" hreflang="bs" data-title="Teorija grafova" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_grafs" title="Teoria de grafs – Catalan" lang="ca" hreflang="ca" data-title="Teoria de grafs" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D1%84%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Графсен теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Графсен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_graf%C5%AF" title="Teorie grafů – Czech" lang="cs" hreflang="cs" data-title="Teorie grafů" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_graffiau" title="Damcaniaeth graffiau – Welsh" lang="cy" hreflang="cy" data-title="Damcaniaeth graffiau" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Grafteori" title="Grafteori – Danish" lang="da" hreflang="da" data-title="Grafteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Graphentheorie" title="Graphentheorie – German" lang="de" hreflang="de" data-title="Graphentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Graafiteooria" title="Graafiteooria – Estonian" lang="et" hreflang="et" data-title="Graafiteooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B3%CF%81%CE%AC%CF%86%CF%89%CE%BD" title="Θεωρία γράφων – Greek" lang="el" hreflang="el" data-title="Θεωρία γράφων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos – Spanish" lang="es" hreflang="es" data-title="Teoría de grafos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grafeoteorio" title="Grafeoteorio – Esperanto" lang="eo" hreflang="eo" data-title="Grafeoteorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Grafo_teoria" title="Grafo teoria – Basque" lang="eu" hreflang="eu" data-title="Grafo teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D8%A7%D9%81" title="نظریه گراف – Persian" lang="fa" hreflang="fa" data-title="نظریه گراف" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_graphes" title="Théorie des graphes – French" lang="fr" hreflang="fr" data-title="Théorie des graphes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_grafos" title="Teoría de grafos – Galician" lang="gl" hreflang="gl" data-title="Teoría de grafos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B8%EB%9E%98%ED%94%84_%EC%9D%B4%EB%A1%A0" title="그래프 이론 – Korean" lang="ko" hreflang="ko" data-title="그래프 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D6%80%D5%A1%D6%86%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Գրաֆների տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Գրաֆների տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AB%E0%A4%BC_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="ग्राफ़ सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="ग्राफ़ सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_grafova" title="Teorija grafova – Croatian" lang="hr" hreflang="hr" data-title="Teorija grafova" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Grafikoteorio" title="Grafikoteorio – Ido" lang="io" hreflang="io" data-title="Grafikoteorio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_graf" title="Teori graf – Indonesian" lang="id" hreflang="id" data-title="Teori graf" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Netafr%C3%A6%C3%B0i" title="Netafræði – Icelandic" lang="is" hreflang="is" data-title="Netafræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dei_grafi" title="Teoria dei grafi – Italian" lang="it" hreflang="it" data-title="Teoria dei grafi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%92%D7%A8%D7%A4%D7%99%D7%9D" title="תורת הגרפים – Hebrew" lang="he" hreflang="he" data-title="תורת הגרפים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%A0%E1%83%90%E1%83%A4%E1%83%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="გრაფთა თეორია – Georgian" lang="ka" hreflang="ka" data-title="გრაფთა თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D1%84%D1%82%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Графтар теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Графтар теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D1%84%D1%82%D0%B0%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Графтар теориясы – Kyrgyz" lang="ky" hreflang="ky" data-title="Графтар теориясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grafu_teorija" title="Grafu teorija – Latvian" lang="lv" hreflang="lv" data-title="Grafu teorija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Graf%C5%B3_teorija" title="Grafų teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Grafų teorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gr%C3%A1felm%C3%A9let" title="Gráfelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Gráfelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D1%84%D0%BE%D0%B2%D0%B8" title="Теорија на графови – Macedonian" lang="mk" hreflang="mk" data-title="Теорија на графови" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Teorija_tal-grafi" title="Teorija tal-grafi – Maltese" lang="mt" hreflang="mt" data-title="Teorija tal-grafi" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_graf" title="Teori graf – Malay" lang="ms" hreflang="ms" data-title="Teori graf" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D1%84%D1%8B%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB" title="Графын онол – Mongolian" lang="mn" hreflang="mn" data-title="Графын онол" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Grafentheorie" title="Grafentheorie – Dutch" lang="nl" hreflang="nl" data-title="Grafentheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B0%E3%83%A9%E3%83%95%E7%90%86%E8%AB%96" title="グラフ理論 – Japanese" lang="ja" hreflang="ja" data-title="グラフ理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Grafteori" title="Grafteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Grafteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Grafteori" title="Grafteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Grafteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_graf%C3%B3w" title="Teoria grafów – Polish" lang="pl" hreflang="pl" data-title="Teoria grafów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_dos_grafos" title="Teoria dos grafos – Portuguese" lang="pt" hreflang="pt" data-title="Teoria dos grafos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_grafurilor" title="Teoria grafurilor – Romanian" lang="ro" hreflang="ro" data-title="Teoria grafurilor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B3%D1%80%D0%B0%D1%84%D0%BE%D0%B2" title="Теория графов – Russian" lang="ru" hreflang="ru" data-title="Теория графов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_grafeve" title="Teoria e grafeve – Albanian" lang="sq" hreflang="sq" data-title="Teoria e grafeve" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiuria_d%C3%AE_grafi" title="Tiuria dî grafi – Sicilian" lang="scn" hreflang="scn" data-title="Tiuria dî grafi" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Graph_theory" title="Graph theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Graph theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_grafov" title="Teória grafov – Slovak" lang="sk" hreflang="sk" data-title="Teória grafov" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_grafov" title="Teorija grafov – Slovenian" lang="sl" hreflang="sl" data-title="Teorija grafov" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%DA%AF%D8%B1%D8%A7%D9%81" title="تیۆریی گراف – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆریی گراف" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B3%D1%80%D0%B0%D1%84%D0%BE%D0%B2%D0%B0" title="Теорија графова – Serbian" lang="sr" hreflang="sr" data-title="Теорија графова" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_grafova" title="Teorija grafova – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Teorija grafova" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Verkkoteoria" title="Verkkoteoria – Finnish" lang="fi" hreflang="fi" data-title="Verkkoteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Grafteori" title="Grafteori – Swedish" lang="sv" hreflang="sv" data-title="Grafteori" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_grap" title="Teorya ng grap – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng grap" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AF%81%E0%AE%B0%E0%AF%81%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="கோட்டுருவியல் – Tamil" lang="ta" hreflang="ta" data-title="கோட்டுருவியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%81%E0%B8%A3%E0%B8%B2%E0%B8%9F" title="ทฤษฎีกราฟ – Thai" lang="th" hreflang="th" data-title="ทฤษฎีกราฟ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B7%D0%B0%D1%80%D0%B8%D1%8F%D0%B8_%D0%B3%D1%80%D0%B0%D1%84%D2%B3%D0%BE" title="Назарияи графҳо – Tajik" lang="tg" hreflang="tg" data-title="Назарияи графҳо" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87izge_teorisi" title="Çizge teorisi – Turkish" lang="tr" hreflang="tr" data-title="Çizge teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B3%D1%80%D0%B0%D1%84%D1%96%D0%B2" title="Теорія графів – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія графів" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%DA%AF%D8%B1%D8%A7%D9%81" title="نظریۂ گراف – Urdu" lang="ur" hreflang="ur" data-title="نظریۂ گراف" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_%C4%91%E1%BB%93_th%E1%BB%8B" title="Lý thuyết đồ thị – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết đồ thị" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about sets of vertices connected by edges. For graphs of mathematical functions, see <a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a>. For other uses, see <a href="/wiki/Graph_(disambiguation)" class="mw-redirect mw-disambig" title="Graph (disambiguation)">Graph (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Area of discrete mathematics</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_simple_undirected_graph_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Example_of_simple_undirected_graph_3.svg/150px-Example_of_simple_undirected_graph_3.svg.png" decoding="async" width="150" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Example_of_simple_undirected_graph_3.svg/225px-Example_of_simple_undirected_graph_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Example_of_simple_undirected_graph_3.svg/300px-Example_of_simple_undirected_graph_3.svg.png 2x" data-file-width="97" data-file-height="64" /></a><figcaption>A <a href="/wiki/Graph_drawing" title="Graph drawing">drawing</a> of a graph with 6 vertices and 7 edges.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <b>graph theory</b> is the study of <i><a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a></i>, which are <a href="/wiki/Mathematical_structures" class="mw-redirect" title="Mathematical structures">mathematical structures</a> used to model pairwise relations between objects. A graph in this context is made up of <i><a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a></i> (also called <i>nodes</i> or <i>points</i>) which are connected by <i><a href="/wiki/Glossary_of_graph_theory_terms#edge" class="mw-redirect" title="Glossary of graph theory terms">edges</a></i> (also called <i>arcs</i>, <i>links</i> or <i>lines</i>). A distinction is made between <b>undirected graphs</b>, where edges link two vertices symmetrically, and <b>directed graphs</b>, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Glossary_of_graph_theory" title="Glossary of graph theory">Glossary of graph theory</a></div> <p>Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Graph">Graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=2" title="Edit section: Graph"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_simple_undirected_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Example_of_simple_undirected_graph.svg/150px-Example_of_simple_undirected_graph.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Example_of_simple_undirected_graph.svg/225px-Example_of_simple_undirected_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Example_of_simple_undirected_graph.svg/300px-Example_of_simple_undirected_graph.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption>A graph with three vertices and three edges.</figcaption></figure> <p>In one restricted but very common sense of the term,<sup id="cite_ref-FOOTNOTEBenderWilliamson2010148_1-0" class="reference"><a href="#cite_note-FOOTNOTEBenderWilliamson2010148-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> a <b>graph</b> is an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(V,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644a8d85ee410b6159ca2bdb5dcb9097e2c8f182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle G=(V,E)}"></span> comprising: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <b>vertices</b> (also called <b>nodes</b> or <b>points</b>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e9954c722145a63f937b26123915b546cb9db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.749ex; height:2.843ex;" alt="{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <b>edges</b> (also called <b>links</b> or <b>lines</b>), which are <a href="/wiki/Unordered_pair" title="Unordered pair">unordered pairs</a> of vertices (that is, an edge is associated with two distinct vertices).</li></ul> <p>To avoid ambiguity, this type of object may be called precisely an <b>undirected simple graph</b>. </p><p>In the edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2611cdc8fecaffa28cb0ea888dbba55f3a31077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.844ex; height:2.843ex;" alt="{\displaystyle \{x,y\}}"></span>, the vertices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are called the <b>endpoints</b> of the edge. The edge is said to <b>join</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and to be <b>incident</b> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. A vertex may exist in a graph and not belong to an edge. Under this definition, <b><a href="/wiki/Multiple_edges" title="Multiple edges">multiple edges</a></b>, in which two or more edges connect the same vertices, are not allowed. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_simple_undirected_graph_with_loops.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Example_of_simple_undirected_graph_with_loops.svg/150px-Example_of_simple_undirected_graph_with_loops.svg.png" decoding="async" width="150" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Example_of_simple_undirected_graph_with_loops.svg/225px-Example_of_simple_undirected_graph_with_loops.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Example_of_simple_undirected_graph_with_loops.svg/300px-Example_of_simple_undirected_graph_with_loops.svg.png 2x" data-file-width="114" data-file-height="141" /></a><figcaption>Example of simple undirected graph with 3 vertices, 3 edges and 4 loops.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:158px;max-width:158px"><div class="trow"><div class="tsingle" style="width:77px;max-width:77px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Example_of_simple_undirected_graph_2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Example_of_simple_undirected_graph_2.svg/75px-Example_of_simple_undirected_graph_2.svg.png" decoding="async" width="75" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Example_of_simple_undirected_graph_2.svg/113px-Example_of_simple_undirected_graph_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Example_of_simple_undirected_graph_2.svg/150px-Example_of_simple_undirected_graph_2.svg.png 2x" data-file-width="105" data-file-height="107" /></a></span></div><div class="thumbcaption">For vertices A,B,C and D, the degrees are respectively 4,4,5,1</div></div><div class="tsingle" style="width:77px;max-width:77px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Example_of_simple_undirected_graph_1.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Example_of_simple_undirected_graph_1.svg/75px-Example_of_simple_undirected_graph_1.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Example_of_simple_undirected_graph_1.svg/113px-Example_of_simple_undirected_graph_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Example_of_simple_undirected_graph_1.svg/150px-Example_of_simple_undirected_graph_1.svg.png 2x" data-file-width="100" data-file-height="100" /></a></span></div><div class="thumbcaption">For vertices U,V,W and X, the degrees are 2,2,3 and 1 respectively.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Examples of finding the degree of vertices.</div></div></div></div> <p>In one more general sense of the term allowing multiple edges,<sup id="cite_ref-FOOTNOTEBenderWilliamson2010149_3-0" class="reference"><a href="#cite_note-FOOTNOTEBenderWilliamson2010149-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> a <b>graph</b> is an ordered triple <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(V,E,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d427ef20e7ca460e1a8fc6069aa44aa43447c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.751ex; height:2.843ex;" alt="{\displaystyle G=(V,E,\phi )}"></span> comprising: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <b>vertices</b> (also called <b>nodes</b> or <b>points</b>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <b>edges</b> (also called <b>links</b> or <b>lines</b>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d1de96c66b47f9e65d46b28d8a2d6c42927bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.587ex; height:2.843ex;" alt="{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"></span>, an <b>incidence function</b> mapping every edge to an <a href="/wiki/Unordered_pair" title="Unordered pair">unordered pair</a> of vertices (that is, an edge is associated with two distinct vertices).</li></ul> <p>To avoid ambiguity, this type of object may be called precisely an <b>undirected <a href="/wiki/Multigraph" title="Multigraph">multigraph</a></b>. </p><p>A <b><a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">loop</a></b> is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,x\}=\{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,x\}=\{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a457ee3880442411f71879595a0e99c666601791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.771ex; height:2.843ex;" alt="{\displaystyle \{x,x\}=\{x\}}"></span> which is not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70ae8296685423e58fb58f1b7d5061f64d66fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.875ex; height:2.843ex;" alt="{\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}"></span>. To allow loops, the definitions must be expanded. For undirected simple graphs, the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> should be modified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8153f5127d2ca8bce373d2520c36b3fdbb6bff25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.127ex; height:2.843ex;" alt="{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}"></span>. For undirected multigraphs, the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> should be modified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49004a1fd53258a96c0c8d2609fcf83620a1f282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.965ex; height:2.843ex;" alt="{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}"></span>. To avoid ambiguity, these types of objects may be called <b>undirected simple graph permitting loops</b> and <b>undirected multigraph permitting loops</b> (sometimes also <b>undirected <a href="/wiki/Pseudograph" class="mw-redirect" title="Pseudograph">pseudograph</a></b>), respectively. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the <a href="/wiki/Infinite_graph" class="mw-redirect" title="Infinite graph">infinite case</a>. Moreover, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is often assumed to be non-empty, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is allowed to be the empty set. The <b>order</b> of a graph is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddcffc28643ac01a14dd0fb32c3157859e365a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.081ex; height:2.843ex;" alt="{\displaystyle |V|}"></span>, its number of vertices. The <b>size</b> of a graph is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c2b9637808cf805d411190b4ae017dbd4ef8d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.069ex; height:2.843ex;" alt="{\displaystyle |E|}"></span>, its number of edges. The <b>degree</b> or <b>valency</b> of a vertex is the number of edges that are incident to it, where a loop is counted twice. The <b>degree</b> of a graph is the maximum of the degrees of its vertices. </p><p>In an undirected simple graph of order <i>n</i>, the maximum degree of each vertex is <span class="nowrap"><i>n</i> − 1</span> and the maximum size of the graph is <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i>(<i>n</i> − 1)</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. </p><p>The edges of an undirected simple graph permitting loops <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> induce a symmetric <a href="/wiki/Binary_relation#Homogeneous_relation" title="Binary relation">homogeneous relation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> on the vertices of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> that is called the <b>adjacency relation</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Specifically, for each edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>, its endpoints <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are said to be <b>adjacent</b> to one another, which is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd1014d850b7c883eb76301dd58c643e3c7e4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Directed_graph">Directed graph</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=3" title="Edit section: Directed graph"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Directed_graph" title="Directed graph">Directed graph</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_simple_directed_graph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_simple_directed_graph.svg/150px-Example_of_simple_directed_graph.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_simple_directed_graph.svg/225px-Example_of_simple_directed_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Example_of_simple_directed_graph.svg/300px-Example_of_simple_directed_graph.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption>A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).</figcaption></figure> <p>A <b>directed graph</b> or <b>digraph</b> is a graph in which edges have orientations. </p><p>In one restricted but very common sense of the term,<sup id="cite_ref-FOOTNOTEBenderWilliamson2010161_5-0" class="reference"><a href="#cite_note-FOOTNOTEBenderWilliamson2010161-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> a <b>directed graph</b> is an ordered pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(V,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644a8d85ee410b6159ca2bdb5dcb9097e2c8f182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.331ex; height:2.843ex;" alt="{\displaystyle G=(V,E)}"></span> comprising: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <i>vertices</i> (also called <i>nodes</i> or <i>points</i>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d5017f509f81427d4f5bb82b1964dd48cbd583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.612ex; height:3.343ex;" alt="{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <i>edges</i> (also called <i>directed edges</i>, <i>directed links</i>, <i>directed lines</i>, <i>arrows</i> or <i>arcs</i>) which are <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of vertices (that is, an edge is associated with two distinct vertices).</li></ul> <p>To avoid ambiguity, this type of object may be called precisely a <b>directed simple graph</b>. In set theory and graph theory, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbfbf91428ae481b792337be15fdae34db6331ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.135ex; height:2.343ex;" alt="{\displaystyle V^{n}}"></span> denotes the set of <span class="texhtml mvar" style="font-style:italic;">n</span>-<a href="/wiki/Tuple" title="Tuple">tuples</a> of elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> that is, ordered sequences of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> elements that are not necessarily distinct. </p><p>In the edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> directed from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, the vertices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are called the <i>endpoints</i> of the edge, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> the <i>tail</i> of the edge and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> the <i>head</i> of the edge. The edge is said to <i>join</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and to be <i>incident</i> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. A vertex may exist in a graph and not belong to an edge. The edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec736777360ba7cbdabf050bc448d33ec5e266b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (y,x)}"></span> is called the <i>inverted edge</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>. <i><a href="/wiki/Multiple_edges" title="Multiple edges">Multiple edges</a></i>, not allowed under the definition above, are two or more edges with both the same tail and the same head. </p><p>In one more general sense of the term allowing multiple edges,<sup id="cite_ref-FOOTNOTEBenderWilliamson2010161_5-1" class="reference"><a href="#cite_note-FOOTNOTEBenderWilliamson2010161-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> a <b>directed graph</b> is an ordered triple <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(V,E,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(V,E,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d427ef20e7ca460e1a8fc6069aa44aa43447c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.751ex; height:2.843ex;" alt="{\displaystyle G=(V,E,\phi )}"></span> comprising: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <i>vertices</i> (also called <i>nodes</i> or <i>points</i>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <i>edges</i> (also called <i>directed edges</i>, <i>directed links</i>, <i>directed lines</i>, <i>arrows</i> or <i>arcs</i>);</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa82c43bd4a851ec4fedd13d870336488bb9d413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.45ex; height:3.343ex;" alt="{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"></span>, an <i>incidence function</i> mapping every edge to an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of vertices (that is, an edge is associated with two distinct vertices).</li></ul> <p>To avoid ambiguity, this type of object may be called precisely a <b>directed multigraph</b>. </p><p>A <i><a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">loop</a></i> is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f9e25892f6d000349b8bb6578a59567efbdd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle (x,x)}"></span> which is not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25410d40346861db34ad7c20585306c2e0642af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.738ex; height:3.343ex;" alt="{\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}"></span>. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> should be modified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4480299c49fd60e3eaa82f8771bfdff5361f4eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.99ex; height:3.343ex;" alt="{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}"></span>. For directed multigraphs, the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> should be modified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f448ad13540d90792adb6dccdd09210f308aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.829ex; height:3.343ex;" alt="{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}"></span>. To avoid ambiguity, these types of objects may be called precisely a <b>directed simple graph permitting loops</b> and a <b>directed multigraph permitting loops</b> (or a <i><a href="/wiki/Quiver_(mathematics)" title="Quiver (mathematics)">quiver</a></i>) respectively. </p><p>The edges of a directed simple graph permitting loops <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is a <a href="/wiki/Binary_relation#Homogeneous_relation" title="Binary relation">homogeneous relation</a> ~ on the vertices of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> that is called the <i>adjacency relation</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Specifically, for each edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>, its endpoints <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are said to be <i>adjacent</i> to one another, which is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ~ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wikipedia_multilingual_network_graph_July_2013.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wikipedia_multilingual_network_graph_July_2013.svg/220px-Wikipedia_multilingual_network_graph_July_2013.svg.png" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wikipedia_multilingual_network_graph_July_2013.svg/330px-Wikipedia_multilingual_network_graph_July_2013.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Wikipedia_multilingual_network_graph_July_2013.svg/440px-Wikipedia_multilingual_network_graph_July_2013.svg.png 2x" data-file-width="918" data-file-height="841" /></a><figcaption>The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (vertices) during one month in summer 2013.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Graphs can be used to model many types of relations and processes in physical, biological,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> social and information systems.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term <i>network</i> is sometimes defined to mean a graph in which attributes (e.g. names) are associated with the vertices and edges, and the subject that expresses and understands real-world systems as a network is called <a href="/wiki/Network_science" title="Network science">network science</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=5" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Within <a href="/wiki/Computer_science" title="Computer science">computer science</a>, '<a href="/wiki/Cybernetics" title="Cybernetics">causal</a>' and 'non-causal' linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a <a href="/wiki/Website" title="Website">website</a> can be represented by a directed graph, in which the vertices represent web pages and directed edges represent <a href="/wiki/Hyperlink" title="Hyperlink">links</a> from one page to another. A similar approach can be taken to problems in social media,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and many other fields. The development of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> to handle graphs is therefore of major interest in computer science. The <a href="/wiki/Graph_transformation" class="mw-redirect" title="Graph transformation">transformation of graphs</a> is often formalized and represented by <a href="/wiki/Graph_rewriting" title="Graph rewriting">graph rewrite systems</a>. Complementary to <a href="/wiki/Graph_transformation" class="mw-redirect" title="Graph transformation">graph transformation</a> systems focusing on rule-based in-memory manipulation of graphs are <a href="/wiki/Graph_database" title="Graph database">graph databases</a> geared towards <a href="/wiki/Database_transaction" title="Database transaction">transaction</a>-safe, <a href="/wiki/Persistence_(computer_science)" title="Persistence (computer science)">persistent</a> storing and querying of <a href="/wiki/Graph_(data_structure)" class="mw-redirect" title="Graph (data structure)">graph-structured data</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Linguistics">Linguistics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=6" title="Edit section: Linguistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Graph-theoretic methods, in various forms, have proven particularly useful in <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, since natural language often lends itself well to discrete structure. Traditionally, <a href="/wiki/Syntax" title="Syntax">syntax</a> and compositional semantics follow tree-based structures, whose expressive power lies in the <a href="/wiki/Principle_of_compositionality" title="Principle of compositionality">principle of compositionality</a>, modeled in a hierarchical graph. More contemporary approaches such as <a href="/wiki/Head-driven_phrase_structure_grammar" title="Head-driven phrase structure grammar">head-driven phrase structure grammar</a> model the syntax of natural language using <a href="/wiki/Feature_structure" title="Feature structure">typed feature structures</a>, which are <a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graphs</a>. Within <a href="/wiki/Lexical_semantics" title="Lexical semantics">lexical semantics</a>, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; <a href="/wiki/Semantic_network" title="Semantic network">semantic networks</a> are therefore important in <a href="/wiki/Computational_linguistics" title="Computational linguistics">computational linguistics</a>. Still, other methods in phonology (e.g. <a href="/wiki/Optimality_theory" title="Optimality theory">optimality theory</a>, which uses <a href="/wiki/Lattice_graph" title="Lattice graph">lattice graphs</a>) and morphology (e.g. finite-state morphology, using <a href="/wiki/Finite-state_transducer" title="Finite-state transducer">finite-state transducers</a>) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as <a rel="nofollow" class="external text" href="http://www.textgraphs.org/">TextGraphs</a>, as well as various 'Net' projects, such as <a href="/wiki/WordNet" title="WordNet">WordNet</a>, <a href="/wiki/VerbNet" title="VerbNet">VerbNet</a>, and others. </p> <div class="mw-heading mw-heading3"><h3 id="Physics_and_chemistry">Physics and chemistry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=7" title="Edit section: Physics and chemistry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Graph theory is also used to study molecules in <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> and <a href="/wiki/Physics" title="Physics">physics</a>. In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman graphs and rules of calculation</a> summarize <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> in a form in close contact with the experimental numbers one wants to understand."<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In chemistry a graph makes a natural model for a molecule, where vertices represent <a href="/wiki/Atom" title="Atom">atoms</a> and edges <a href="/wiki/Chemical_bond" title="Chemical bond">bonds</a>. This approach is especially used in computer processing of molecular structures, ranging from <a href="/wiki/Molecule_editor" title="Molecule editor">chemical editors</a> to database searching. In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Similarly, in <a href="/wiki/Computational_neuroscience" title="Computational neuroscience">computational neuroscience</a> graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Graphs are also used to represent the micro-scale channels of <a href="/wiki/Porous_medium" title="Porous medium">porous media</a>, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores. <a href="/wiki/Chemical_graph_theory" title="Chemical graph theory">Chemical graph theory</a> uses the <a href="/wiki/Molecular_graph" title="Molecular graph">molecular graph</a> as a means to model molecules. Graphs and networks are excellent models to study and understand phase transitions and critical phenomena. Removal of nodes or edges leads to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via <a href="/wiki/Percolation_theory" title="Percolation theory">percolation theory</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Social_sciences">Social sciences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=8" title="Edit section: Social sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Moreno_Sociogram_2nd_Grade.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Moreno_Sociogram_2nd_Grade.png/220px-Moreno_Sociogram_2nd_Grade.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Moreno_Sociogram_2nd_Grade.png/330px-Moreno_Sociogram_2nd_Grade.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Moreno_Sociogram_2nd_Grade.png/440px-Moreno_Sociogram_2nd_Grade.png 2x" data-file-width="525" data-file-height="525" /></a><figcaption>Graph theory in sociology: <a href="/wiki/Jacob_L._Moreno" title="Jacob L. Moreno">Moreno</a> <a href="/wiki/Sociogram" title="Sociogram">Sociogram</a> (1953).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Graph theory is also widely used in <a href="/wiki/Sociology" title="Sociology">sociology</a> as a way, for example, to <a href="/wiki/Six_Degrees_of_Kevin_Bacon" title="Six Degrees of Kevin Bacon">measure actors' prestige</a> or to explore <a href="/wiki/Rumor_spread_in_social_network" title="Rumor spread in social network">rumor spreading</a>, notably through the use of <a href="/wiki/Social_network_analysis" title="Social network analysis">social network analysis</a> software. Under the umbrella of social networks are many different types of graphs.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together. </p> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=9" title="Edit section: Biology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Likewise, graph theory is useful in <a href="/wiki/Biology" title="Biology">biology</a> and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species. </p><p>Graphs are also commonly used in <a href="/wiki/Molecular_biology" title="Molecular biology">molecular biology</a> and <a href="/wiki/Genomics" title="Genomics">genomics</a> to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in <a href="/wiki/Single-cell_analysis#Transcriptomics" title="Single-cell analysis">single-cell transcriptome analysis</a>. Another use is to model genes or proteins in a <a href="/wiki/Biological_pathway" title="Biological pathway">pathway</a> and study the relationships between them, such as metabolic pathways and gene regulatory networks.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures. </p><p>Graph theory is also used in <a href="/wiki/Connectomics" title="Connectomics">connectomics</a>;<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=10" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematics, graphs are useful in geometry and certain parts of topology such as <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>. <a href="/wiki/Algebraic_graph_theory" title="Algebraic graph theory">Algebraic graph theory</a> has close links with <a href="/wiki/Group_theory" title="Group theory">group theory</a>. Algebraic graph theory has been applied to many areas including dynamic systems and complexity. </p> <div class="mw-heading mw-heading3"><h3 id="Other_topics">Other topics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=11" title="Edit section: Other topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or <a href="/wiki/Weighted_graph" class="mw-redirect" title="Weighted graph">weighted graphs</a>, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=12" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Konigsberg_bridges.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Konigsberg_bridges.png/220px-Konigsberg_bridges.png" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5d/Konigsberg_bridges.png 1.5x" data-file-width="302" data-file-height="238" /></a><figcaption>The Königsberg Bridge problem</figcaption></figure> <p>The paper written by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> on the <a href="/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title="Seven Bridges of Königsberg">Seven Bridges of Königsberg</a> and published in 1736 is regarded as the first paper in the history of graph theory.<sup id="cite_ref-Biggs_20-0" class="reference"><a href="#cite_note-Biggs-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This paper, as well as the one written by <a href="/wiki/Alexandre-Th%C3%A9ophile_Vandermonde" title="Alexandre-Théophile Vandermonde">Vandermonde</a> on the <i><a href="/wiki/Knight%27s_tour" title="Knight's tour">knight problem</a>,</i> carried on with the <i>analysis situs</i> initiated by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a><sup id="cite_ref-Cauchy_21-0" class="reference"><a href="#cite_note-Cauchy-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Simon_Antoine_Jean_L%27Huilier" title="Simon Antoine Jean L'Huilier">L'Huilier</a>,<sup id="cite_ref-Lhuillier_22-0" class="reference"><a href="#cite_note-Lhuillier-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and represents the beginning of the branch of mathematics known as <a href="/wiki/Topology" title="Topology">topology</a>. </p><p>More than one century after Euler's paper on the bridges of <a href="/wiki/K%C3%B6nigsberg" title="Königsberg">Königsberg</a> and while <a href="/wiki/Johann_Benedict_Listing" title="Johann Benedict Listing">Listing</a> was introducing the concept of topology, <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> was led by an interest in particular analytical forms arising from <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> to study a particular class of graphs, the <i><a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">trees</a></i>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> This study had many implications for theoretical <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>. The techniques he used mainly concern the <a href="/wiki/Enumeration_of_graphs" class="mw-redirect" title="Enumeration of graphs">enumeration of graphs</a> with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by <a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya</a> between 1935 and 1937. These were generalized by <a href="/wiki/Nicolaas_Govert_de_Bruijn" title="Nicolaas Govert de Bruijn">De Bruijn</a> in 1959. Cayley linked his results on trees with contemporary studies of chemical composition.<sup id="cite_ref-Cayley1_24-0" class="reference"><a href="#cite_note-Cayley1-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory. </p><p>In particular, the term "graph" was introduced by <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">Sylvester</a> in a paper published in 1878 in <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:<sup id="cite_ref-Sylvester_25-0" class="reference"><a href="#cite_note-Sylvester-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>"[…] Every invariant and co-variant thus becomes expressible by a <i>graph</i> precisely identical with a <a href="/wiki/August_Kekul%C3%A9" title="August Kekulé">Kekuléan</a> diagram or chemicograph. […] I give a rule for the geometrical multiplication of graphs, <i>i.e.</i> for constructing a <i>graph</i> to the product of in- or co-variants whose separate graphs are given. […]" (italics as in the original).</dd></dl> <p>The first textbook on graph theory was written by <a href="/wiki/D%C3%A9nes_K%C5%91nig" title="Dénes Kőnig">Dénes Kőnig</a>, and published in 1936.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Another book by <a href="/wiki/Frank_Harary" title="Frank Harary">Frank Harary</a>, published in 1969, was "considered the world over to be the definitive textbook on the subject",<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other. Harary donated all of the royalties to fund the <a href="/wiki/George_P%C3%B3lya_Prize" title="George Pólya Prize">Pólya Prize</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>One of the most famous and stimulating problems in graph theory is the <a href="/wiki/Four_color_problem" class="mw-redirect" title="Four color problem">four color problem</a>: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by <a href="/wiki/Francis_Guthrie" title="Francis Guthrie">Francis Guthrie</a> in 1852 and its first written record is in a letter of <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a> addressed to <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> the same year. Many incorrect proofs have been proposed, including those by Cayley, <a href="/wiki/Alfred_Kempe" title="Alfred Kempe">Kempe</a>, and others. The study and the generalization of this problem by <a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a>, <a href="/wiki/Percy_John_Heawood" title="Percy John Heawood">Heawood</a>, <a href="/wiki/Frank_P._Ramsey" class="mw-redirect" title="Frank P. Ramsey">Ramsey</a> and <a href="/wiki/Hugo_Hadwiger" title="Hugo Hadwiger">Hadwiger</a> led to the study of the colorings of the graphs embedded on surfaces with arbitrary <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a>. Tait's reformulation generated a new class of problems, the <i>factorization problems</i>, particularly studied by <a href="/wiki/Julius_Petersen" title="Julius Petersen">Petersen</a> and <a href="/wiki/D%C3%A9nes_K%C5%91nig" title="Dénes Kőnig">Kőnig</a>. The works of Ramsey on colorations and more specially the results obtained by <a href="/wiki/P%C3%A1l_Tur%C3%A1n" title="Pál Turán">Turán</a> in 1941 was at the origin of another branch of graph theory, <i><a href="/wiki/Extremal_graph_theory" title="Extremal graph theory">extremal graph theory</a></i>. </p><p>The four color problem remained unsolved for more than a century. In 1969 <a href="/wiki/Heinrich_Heesch" title="Heinrich Heesch">Heinrich Heesch</a> published a method for solving the problem using computers.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> A computer-aided proof produced in 1976 by <a href="/wiki/Kenneth_Appel" title="Kenneth Appel">Kenneth Appel</a> and <a href="/wiki/Wolfgang_Haken" title="Wolfgang Haken">Wolfgang Haken</a> makes fundamental use of the notion of "discharging" developed by Heesch.<sup id="cite_ref-AA1_30-0" class="reference"><a href="#cite_note-AA1-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-AA2_31-0" class="reference"><a href="#cite_note-AA2-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by <a href="/wiki/Neil_Robertson_(mathematician)" title="Neil Robertson (mathematician)">Robertson</a>, <a href="/wiki/Paul_Seymour_(mathematician)" title="Paul Seymour (mathematician)">Seymour</a>, <a href="/wiki/Daniel_P._Sanders" title="Daniel P. Sanders">Sanders</a> and <a href="/wiki/Robin_Thomas_(mathematician)" title="Robin Thomas (mathematician)">Thomas</a>.<sup id="cite_ref-RSST_32-0" class="reference"><a href="#cite_note-RSST-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of <a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan</a>, <a href="/wiki/Kazimierz_Kuratowski" title="Kazimierz Kuratowski">Kuratowski</a> and <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Whitney</a>. Another important factor of common development of graph theory and <a href="/wiki/Topology" title="Topology">topology</a> came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a>, who published in 1845 his <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's circuit laws</a> for calculating the <a href="/wiki/Voltage" title="Voltage">voltage</a> and <a href="/wiki/Electric_current" title="Electric current">current</a> in <a href="/wiki/Electric_circuit" class="mw-redirect" title="Electric circuit">electric circuits</a>. </p><p>The introduction of probabilistic methods in graph theory, especially in the study of <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős</a> and <a href="/wiki/Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Rényi</a> of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as <i><a href="/wiki/Random_graph" title="Random graph">random graph theory</a></i>, which has been a fruitful source of graph-theoretic results. </p> <div class="mw-heading mw-heading2"><h2 id="Representation">Representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=13" title="Edit section: Representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A graph is an abstraction of relationships that emerge in nature; hence, it cannot be coupled to a certain representation. The way it is represented depends on the degree of convenience such representation provides for a certain application. The most common representations are the visual, in which, usually, vertices are drawn and connected by edges, and the tabular, in which rows of a table provide information about the relationships between the vertices within the graph. </p> <div class="mw-heading mw-heading3"><h3 id="Visual:_Graph_drawing">Visual: Graph drawing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=14" title="Edit section: Visual: Graph drawing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></div> <p>Graphs are usually represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. If the graph is weighted, the weight is added on the arrow. </p><p>A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others. </p><p>The pioneering work of <a href="/wiki/W._T._Tutte" title="W. T. Tutte">W. T. Tutte</a> was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings. </p><p>Graph drawing also can be said to encompass problems that deal with the <a href="/wiki/Crossing_number_(graph_theory)" title="Crossing number (graph theory)">crossing number</a> and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a <a href="/wiki/Planar_graph" title="Planar graph">planar graph</a>, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied. </p><p>There are other techniques to visualize a graph away from vertices and edges, including <a href="/wiki/Circle_packing_theorem" title="Circle packing theorem">circle packings</a>, <a href="/wiki/Intersection_graph" title="Intersection graph">intersection graph</a>, and other visualizations of the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Tabular:_Graph_data_structures">Tabular: Graph data structures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=15" title="Edit section: Tabular: Graph data structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph (abstract data type)</a></div> <p>The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The <a href="/wiki/Data_structure" title="Data structure">data structure</a> used depends on both the graph structure and the <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for <a href="/wiki/Sparse_graph" class="mw-redirect" title="Sparse graph">sparse graphs</a> as they have smaller memory requirements. <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>List structures include the <a href="/wiki/Edge_list" title="Edge list">edge list</a>, an array of pairs of vertices, and the <a href="/wiki/Adjacency_list" title="Adjacency list">adjacency list</a>, which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to. </p><p>Matrix structures include the <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrix</a>, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>, in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The <a href="/wiki/Degree_matrix" title="Degree matrix">degree matrix</a> indicates the degree of vertices. The <a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a> is a modified form of the adjacency matrix that incorporates information about the <a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">degrees</a> of the vertices, and is useful in some calculations such as <a href="/wiki/Kirchhoff%27s_theorem" title="Kirchhoff's theorem">Kirchhoff's theorem</a> on the number of <a href="/wiki/Spanning_tree" title="Spanning tree">spanning trees</a> of a graph. The <a href="/wiki/Distance_matrix" title="Distance matrix">distance matrix</a>, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a <a href="/wiki/Shortest_path" class="mw-redirect" title="Shortest path">shortest path</a> between two vertices. </p> <div class="mw-heading mw-heading2"><h2 id="Problems">Problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=16" title="Edit section: Problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Enumeration">Enumeration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=17" title="Edit section: Enumeration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a large literature on <a href="/wiki/Graphical_enumeration" class="mw-redirect" title="Graphical enumeration">graphical enumeration</a>: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973). </p> <div class="mw-heading mw-heading3"><h3 id="Subgraphs,_induced_subgraphs,_and_minors"><span id="Subgraphs.2C_induced_subgraphs.2C_and_minors"></span>Subgraphs, induced subgraphs, and minors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=18" title="Edit section: Subgraphs, induced subgraphs, and minors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A common problem, called the <a href="/wiki/Subgraph_isomorphism_problem" title="Subgraph isomorphism problem">subgraph isomorphism problem</a>, is finding a fixed graph as a <a href="/wiki/Glossary_of_graph_theory#Subgraphs" title="Glossary of graph theory">subgraph</a> in a given graph. One reason to be interested in such a question is that many <a href="/wiki/Graph_properties" class="mw-redirect" title="Graph properties">graph properties</a> are <i>hereditary</i> for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Finding maximal subgraphs of a certain kind is often an <a href="/wiki/NP-complete_problem" class="mw-redirect" title="NP-complete problem">NP-complete problem</a>. For example: </p> <ul><li>Finding the largest complete subgraph is called the <a href="/wiki/Clique_problem" title="Clique problem">clique problem</a> (NP-complete).</li></ul> <p>One special case of subgraph isomorphism is the <a href="/wiki/Graph_isomorphism_problem" title="Graph isomorphism problem">graph isomorphism problem</a>. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time. </p><p>A similar problem is finding <a href="/wiki/Induced_subgraph" title="Induced subgraph">induced subgraphs</a> in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example: </p> <ul><li>Finding the largest edgeless induced subgraph or <a href="/wiki/Independent_set_(graph_theory)" title="Independent set (graph theory)">independent set</a> is called the <a href="/wiki/Independent_set_problem" class="mw-redirect" title="Independent set problem">independent set problem</a> (NP-complete).</li></ul> <p>Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A <a href="/wiki/Minor_(graph_theory)" class="mw-redirect" title="Minor (graph theory)">minor</a> or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, <a href="/wiki/Wagner%27s_theorem" title="Wagner's theorem">Wagner's Theorem</a> states: </p> <ul><li>A graph is <a href="/wiki/Planar_graph" title="Planar graph">planar</a> if it contains as a minor neither the <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a> <i>K</i><sub>3,3</sub> (see the <a href="/wiki/Three-cottage_problem" class="mw-redirect" title="Three-cottage problem">Three-cottage problem</a>) nor the complete graph <i>K</i><sub>5</sub>.</li></ul> <p>A similar problem, the subdivision containment problem, is to find a fixed graph as a <a href="/wiki/Subdivision_(graph_theory)" class="mw-redirect" title="Subdivision (graph theory)">subdivision</a> of a given graph. A <a href="/wiki/Subdivision_(graph_theory)" class="mw-redirect" title="Subdivision (graph theory)">subdivision</a> or <a href="/wiki/Homeomorphism_(graph_theory)" title="Homeomorphism (graph theory)">homeomorphism</a> of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as <a href="/wiki/Planarity_(graph_theory)" class="mw-redirect" title="Planarity (graph theory)">planarity</a>. For example, <a href="/wiki/Kuratowski%27s_theorem" title="Kuratowski's theorem">Kuratowski's Theorem</a> states: </p> <ul><li>A graph is <a href="/wiki/Planar_graph" title="Planar graph">planar</a> if it contains as a subdivision neither the <a href="/wiki/Complete_bipartite_graph" title="Complete bipartite graph">complete bipartite graph</a> <i>K</i><sub>3,3</sub> nor the <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> <i>K</i><sub>5</sub>.</li></ul> <p>Another problem in subdivision containment is the <a href="/wiki/Kelmans%E2%80%93Seymour_conjecture" title="Kelmans–Seymour conjecture">Kelmans–Seymour conjecture</a>: </p> <ul><li>Every <a href="/wiki/K-vertex-connected_graph" title="K-vertex-connected graph">5-vertex-connected</a> graph that is not <a href="/wiki/Planar_graph" title="Planar graph">planar</a> contains a <a href="/wiki/Homeomorphism_(graph_theory)" title="Homeomorphism (graph theory)">subdivision</a> of the 5-vertex <a href="/wiki/Complete_graph" title="Complete graph">complete graph</a> <i>K</i><sub>5</sub>.</li></ul> <p>Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their <i>point-deleted subgraphs</i>. For example: </p> <ul><li>The <a href="/wiki/Reconstruction_conjecture" title="Reconstruction conjecture">reconstruction conjecture</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graph_coloring">Graph coloring</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=19" title="Edit section: Graph coloring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_coloring" title="Graph coloring">Graph coloring</a></div> <p>Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two coincident edges are the same color), or other variations. Among the famous results and conjectures concerning graph coloring are the following: </p> <ul><li><a href="/wiki/Four-color_theorem" class="mw-redirect" title="Four-color theorem">Four-color theorem</a></li> <li><a href="/wiki/Strong_perfect_graph_theorem" title="Strong perfect graph theorem">Strong perfect graph theorem</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture" title="Erdős–Faber–Lovász conjecture">Erdős–Faber–Lovász conjecture</a></li> <li><a href="/wiki/Total_coloring" title="Total coloring">Total coloring conjecture</a>, also called <a href="/wiki/Mehdi_Behzad" title="Mehdi Behzad">Behzad</a>'s conjecture (unsolved)</li> <li><a href="/wiki/List_edge-coloring" title="List edge-coloring">List coloring conjecture</a> (unsolved)</li> <li><a href="/wiki/Hadwiger_conjecture_(graph_theory)" title="Hadwiger conjecture (graph theory)">Hadwiger conjecture (graph theory)</a> (unsolved)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Subsumption_and_unification">Subsumption and unification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=20" title="Edit section: Subsumption and unification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Constraint modeling theories concern families of directed graphs related by a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general. Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known. </p><p>For constraint frameworks which are strictly <a href="/wiki/Principle_of_Compositionality" class="mw-redirect" title="Principle of Compositionality">compositional</a>, graph unification is the sufficient satisfiability and combination function. Well-known applications include <a href="/wiki/Automatic_theorem_prover" class="mw-redirect" title="Automatic theorem prover">automatic theorem proving</a> and modeling the <a href="/wiki/Parsing" title="Parsing">elaboration of linguistic structure</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Route_problems">Route problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=21" title="Edit section: Route problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hamiltonian_path_problem" title="Hamiltonian path problem">Hamiltonian path problem</a></li> <li><a href="/wiki/Minimum_spanning_tree" title="Minimum spanning tree">Minimum spanning tree</a></li> <li><a href="/wiki/Route_inspection_problem" class="mw-redirect" title="Route inspection problem">Route inspection problem</a> (also called the "Chinese postman problem")</li> <li><a href="/wiki/Seven_bridges_of_K%C3%B6nigsberg" class="mw-redirect" title="Seven bridges of Königsberg">Seven bridges of Königsberg</a></li> <li><a href="/wiki/Shortest_path_problem" title="Shortest path problem">Shortest path problem</a></li> <li><a href="/wiki/Steiner_tree" class="mw-redirect" title="Steiner tree">Steiner tree</a></li> <li><a href="/wiki/Three-cottage_problem" class="mw-redirect" title="Three-cottage problem">Three-cottage problem</a></li> <li><a href="/wiki/Traveling_salesman_problem" class="mw-redirect" title="Traveling salesman problem">Traveling salesman problem</a> (NP-hard)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Network_flow">Network flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=22" title="Edit section: Network flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are numerous problems arising especially from applications that have to do with various notions of <a href="/wiki/Flow_network" title="Flow network">flows in networks</a>, for example: </p> <ul><li><a href="/wiki/Max_flow_min_cut_theorem" class="mw-redirect" title="Max flow min cut theorem">Max flow min cut theorem</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Visibility_problems">Visibility problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=23" title="Edit section: Visibility problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Museum_guard_problem" class="mw-redirect" title="Museum guard problem">Museum guard problem</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Covering_problems">Covering problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=24" title="Edit section: Covering problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Covering_problem" class="mw-redirect" title="Covering problem">Covering problems</a> in graphs may refer to various <a href="/wiki/Set_cover_problem" title="Set cover problem"> set cover problems</a> on subsets of vertices/subgraphs. </p> <ul><li><a href="/wiki/Dominating_set" title="Dominating set">Dominating set</a> problem is the special case of set cover problem where sets are the closed <a href="/wiki/Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">neighborhoods</a>.</li> <li><a href="/wiki/Vertex_cover_problem" class="mw-redirect" title="Vertex cover problem">Vertex cover problem</a> is the special case of set cover problem where sets to cover are every edges.</li> <li>The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Decomposition_problems">Decomposition problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=25" title="Edit section: Decomposition problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of questions. Often, the problem is to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph <i>K</i><sub><i>n</i></sub> into <span class="nowrap"><i>n</i> − 1</span> specified trees having, respectively, 1, 2, 3, ..., <span class="nowrap"><i>n</i> − 1</span> edges. </p><p>Some specific decomposition problems that have been studied include: </p> <ul><li><a href="/wiki/Arboricity" title="Arboricity">Arboricity</a>, a decomposition into as few forests as possible</li> <li><a href="/wiki/Cycle_double_cover" title="Cycle double cover">Cycle double cover</a>, a decomposition into a collection of cycles covering each edge exactly twice</li> <li><a href="/wiki/Edge_coloring" title="Edge coloring">Edge coloring</a>, a decomposition into as few <a href="/wiki/Matching_(graph_theory)" title="Matching (graph theory)">matchings</a> as possible</li> <li><a href="/wiki/Graph_factorization" title="Graph factorization">Graph factorization</a>, a decomposition of a <a href="/wiki/Regular_graph" title="Regular graph">regular graph</a> into regular subgraphs of given degrees</li></ul> <div class="mw-heading mw-heading3"><h3 id="Graph_classes">Graph classes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=26" title="Edit section: Graph classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below: </p> <ul><li><a href="/wiki/Graph_enumeration" title="Graph enumeration">Enumerating</a> the members of a class</li> <li>Characterizing a class in terms of <a href="/wiki/Forbidden_graph_characterization" title="Forbidden graph characterization">forbidden substructures</a></li> <li>Ascertaining relationships among classes (e.g. does one property of graphs imply another)</li> <li>Finding efficient <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> to <a href="/wiki/Decision_problem" title="Decision problem">decide</a> membership in a class</li> <li>Finding <a href="/wiki/Representation_(mathematics)" title="Representation (mathematics)">representations</a> for members of a class</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Gallery_of_named_graphs" class="mw-redirect" title="Gallery of named graphs">Gallery of named graphs</a></li> <li><a href="/wiki/Glossary_of_graph_theory" title="Glossary of graph theory">Glossary of graph theory</a></li> <li><a href="/wiki/List_of_graph_theory_topics" title="List of graph theory topics">List of graph theory topics</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_graph_theory" class="mw-redirect" title="List of unsolved problems in graph theory">List of unsolved problems in graph theory</a></li> <li><a href="/wiki/List_of_publications_in_mathematics#Graph_theory" class="mw-redirect" title="List of publications in mathematics">Publications in graph theory</a></li> <li><a href="/wiki/Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph algorithm</a></li> <li><a href="/wiki/Category:Graph_theorists" title="Category:Graph theorists">Graph theorists</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Subareas">Subareas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=28" title="Edit section: Subareas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_graph_theory" title="Algebraic graph theory">Algebraic graph theory</a></li> <li><a href="/wiki/Geometric_graph_theory" title="Geometric graph theory">Geometric graph theory</a></li> <li><a href="/wiki/Extremal_graph_theory" title="Extremal graph theory">Extremal graph theory</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Probabilistic graph theory</a></li> <li><a href="/wiki/Topological_graph_theory" title="Topological graph theory">Topological graph theory</a></li> <li><a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=29" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBenderWilliamson2010148-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBenderWilliamson2010148_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBenderWilliamson2010">Bender & Williamson 2010</a>, p. 148.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">See, for instance, Iyanaga and Kawada, <i>69 J</i>, p. 234 or Biggs, p. 4.</span> </li> <li id="cite_note-FOOTNOTEBenderWilliamson2010149-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBenderWilliamson2010149_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBenderWilliamson2010">Bender & Williamson 2010</a>, p. 149.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">See, for instance, Graham et al., p. 5.</span> </li> <li id="cite_note-FOOTNOTEBenderWilliamson2010161-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBenderWilliamson2010161_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBenderWilliamson2010161_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBenderWilliamson2010">Bender & Williamson 2010</a>, p. 161.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHale2014" 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Math.</i>, <b>21</b> (3): 491–567, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fijm%2F1256049012">10.1215/ijm/1256049012</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Illinois+J.+Math.&rft.atitle=Every+planar+map+is+four+colorable.+Part+II.+Reducibility&rft.volume=21&rft.issue=3&rft.pages=491-567&rft.date=1977&rft_id=info%3Adoi%2F10.1215%2Fijm%2F1256049012&rft.au=Appel%2C+K.&rft.au=Haken%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></span> </li> <li id="cite_note-RSST-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-RSST_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertson,_N.Sanders,_D.Seymour,_P.Thomas,_R.1997" class="citation cs2">Robertson, N.; Sanders, D.; Seymour, P.; Thomas, R. (1997), "The four color theorem", <i>Journal of Combinatorial Theory, Series B</i>, <b>70</b>: 2–44, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjctb.1997.1750">10.1006/jctb.1997.1750</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory%2C+Series+B&rft.atitle=The+four+color+theorem&rft.volume=70&rft.pages=2-44&rft.date=1997&rft_id=info%3Adoi%2F10.1006%2Fjctb.1997.1750&rft.au=Robertson%2C+N.&rft.au=Sanders%2C+D.&rft.au=Seymour%2C+P.&rft.au=Thomas%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKepnerGilbert2011" class="citation book cs1">Kepner, Jeremy; Gilbert, John (2011). <a rel="nofollow" class="external text" href="https://my.siam.org/Store/Product/viewproduct/?ProductId=106663"><i>Graph Algorithms in the Language of Linear Algebra</i></a>. SIAM. p. 1171458. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-898719-90-1" title="Special:BookSources/978-0-898719-90-1"><bdi>978-0-898719-90-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Algorithms+in+the+Language+of+Linear+Algebra&rft.pages=1171458&rft.pub=SIAM&rft.date=2011&rft.isbn=978-0-898719-90-1&rft.aulast=Kepner&rft.aufirst=Jeremy&rft.au=Gilbert%2C+John&rft_id=https%3A%2F%2Fmy.siam.org%2FStore%2FProduct%2Fviewproduct%2F%3FProductId%3D106663&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenderWilliamson2010" class="citation book cs1">Bender, Edward A.; Williamson, S. Gill (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vaXv_yhefG8C"><i>Lists, Decisions and Graphs. With an Introduction to Probability</i></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lists%2C+Decisions+and+Graphs.+With+an+Introduction+to+Probability&rft.date=2010&rft.aulast=Bender&rft.aufirst=Edward+A.&rft.au=Williamson%2C+S.+Gill&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvaXv_yhefG8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerge1958" class="citation book cs1">Berge, Claude (1958). <i>Théorie des graphes et ses applications</i>. Paris: Dunod.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+graphes+et+ses+applications&rft.place=Paris&rft.pub=Dunod&rft.date=1958&rft.aulast=Berge&rft.aufirst=Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span> English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition, Dover, New York 2001.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggsLloydWilson1986" class="citation book cs1">Biggs, N.; Lloyd, E.; Wilson, R. (1986). <i>Graph Theory, 1736–1936</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Theory%2C+1736%E2%80%931936&rft.pub=Oxford+University+Press&rft.date=1986&rft.aulast=Biggs&rft.aufirst=N.&rft.au=Lloyd%2C+E.&rft.au=Wilson%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBondyMurty2008" class="citation book cs1">Bondy, J. A.; Murty, U. S. R. (2008). <i>Graph Theory</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84628-969-9" title="Special:BookSources/978-1-84628-969-9"><bdi>978-1-84628-969-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Theory&rft.pub=Springer&rft.date=2008&rft.isbn=978-1-84628-969-9&rft.aulast=Bondy&rft.aufirst=J.+A.&rft.au=Murty%2C+U.+S.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBollobásRiordan2003" class="citation book cs1">Bollobás, Béla; Riordan, O. M. (2003). <i>Mathematical results on scale-free random graphs in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds))</i> (1st ed.). Weinheim: Wiley VCH.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+results+on+scale-free+random+graphs+in+%22Handbook+of+Graphs+and+Networks%22+%28S.+Bornholdt+and+H.G.+Schuster+%28eds%29%29&rft.place=Weinheim&rft.edition=1st&rft.pub=Wiley+VCH&rft.date=2003&rft.aulast=Bollob%C3%A1s&rft.aufirst=B%C3%A9la&rft.au=Riordan%2C+O.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChartrand1985" class="citation book cs1">Chartrand, Gary (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductorygrap0000char"><i>Introductory Graph Theory</i></a></span>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-24775-9" title="Special:BookSources/0-486-24775-9"><bdi>0-486-24775-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Graph+Theory&rft.pub=Dover&rft.date=1985&rft.isbn=0-486-24775-9&rft.aulast=Chartrand&rft.aufirst=Gary&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductorygrap0000char&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeo1974" class="citation book cs1">Deo, Narsingh (1974). <a rel="nofollow" class="external text" href="https://www.edutechlearners.com/download/Graphtheory.pdf"><i>Graph Theory with Applications to Engineering and Computer Science</i></a> <span class="cs1-format">(PDF)</span>. Englewood, New Jersey: Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-363473-6" title="Special:BookSources/0-13-363473-6"><bdi>0-13-363473-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190517165158/http://www.edutechlearners.com/download/Graphtheory.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2019-05-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Theory+with+Applications+to+Engineering+and+Computer+Science&rft.place=Englewood%2C+New+Jersey&rft.pub=Prentice-Hall&rft.date=1974&rft.isbn=0-13-363473-6&rft.aulast=Deo&rft.aufirst=Narsingh&rft_id=https%3A%2F%2Fwww.edutechlearners.com%2Fdownload%2FGraphtheory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbons1985" class="citation book cs1">Gibbons, Alan (1985). <i>Algorithmic Graph Theory</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithmic+Graph+Theory&rft.pub=Cambridge+University+Press&rft.date=1985&rft.aulast=Gibbons&rft.aufirst=Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolumbic1980" class="citation book cs1">Golumbic, Martin (1980). <i>Algorithmic Graph Theory and Perfect Graphs</i>. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithmic+Graph+Theory+and+Perfect+Graphs&rft.pub=Academic+Press&rft.date=1980&rft.aulast=Golumbic&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarary1969" class="citation book cs1">Harary, Frank (1969). <i>Graph Theory</i>. Reading, Massachusetts: Addison-Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Theory&rft.place=Reading%2C+Massachusetts&rft.pub=Addison-Wesley&rft.date=1969&rft.aulast=Harary&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHararyPalmer1973" class="citation book cs1">Harary, Frank; Palmer, Edgar M. (1973). <i>Graphical Enumeration</i>. New York, New York: Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graphical+Enumeration&rft.place=New+York%2C+New+York&rft.pub=Academic+Press&rft.date=1973&rft.aulast=Harary&rft.aufirst=Frank&rft.au=Palmer%2C+Edgar+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahadevPeled1995" class="citation book cs1">Mahadev, N. V. R.; Peled, Uri N. (1995). <i>Threshold Graphs and Related Topics</i>. <a href="/wiki/North-Holland_Publishing_Company" class="mw-redirect" title="North-Holland Publishing Company">North-Holland</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Threshold+Graphs+and+Related+Topics&rft.pub=North-Holland&rft.date=1995&rft.aulast=Mahadev&rft.aufirst=N.+V.+R.&rft.au=Peled%2C+Uri+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewman2010" class="citation book cs1">Newman, Mark (2010). <i>Networks: An Introduction</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Networks%3A+An+Introduction&rft.pub=Oxford+University+Press&rft.date=2010&rft.aulast=Newman&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKepnerGilbert2011" class="citation book cs1">Kepner, Jeremy; Gilbert, John (2011). <a rel="nofollow" class="external text" href="https://my.siam.org/Store/Product/viewproduct/?ProductId=106663"><i>Graph Algorithms in The Language of Linear Algebra</i></a>. Philadelphia, Pennsylvania: SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-898719-90-1" title="Special:BookSources/978-0-898719-90-1"><bdi>978-0-898719-90-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graph+Algorithms+in+The+Language+of+Linear+Algebra&rft.place=Philadelphia%2C+Pennsylvania&rft.pub=SIAM&rft.date=2011&rft.isbn=978-0-898719-90-1&rft.aulast=Kepner&rft.aufirst=Jeremy&rft.au=Gilbert%2C+John&rft_id=https%3A%2F%2Fmy.siam.org%2FStore%2FProduct%2Fviewproduct%2F%3FProductId%3D106663&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=31" title="Edit section: External 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Graph_theory" class="extiw" title="commons:Graph theory"><span style="font-style:italic; font-weight:bold;">Graph theory</span></a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Graph_theory">"Graph theory"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Graph+theory&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DGraph_theory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGraph+theory" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.utm.edu/departments/math/graph/">Graph theory tutorial</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120116185332/http://www.utm.edu/departments/math/graph/">Archived</a> 2012-01-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.gfredericks.com/main/sandbox/graphs">A searchable database of small connected graphs</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060206155001/http://www.nd.edu/~networks/gallery.htm">Image gallery: graphs</a> at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (archived February 6, 2006)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161114100939/http://www.babelgraph.org/links.html">Concise, annotated list of graph theory resources for researchers</a></li> <li><a rel="nofollow" class="external text" href="http://www.kde.org/applications/education/rocs/">rocs</a> — a graph theory IDE</li> <li><a rel="nofollow" class="external text" href="http://www.orgnet.com/SocialLifeOfRouters.pdf">The Social Life of Routers</a> — non-technical paper discussing graphs of people and computers</li> <li><a rel="nofollow" class="external text" href="http://graphtheorysoftware.com/">Graph Theory Software</a> — tools to teach and learn graph theory</li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Graph+theory&library=OLBP">Online books</a>, and library resources <a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Graph+theory">in your library</a> and <a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Graph+theory&library=0CHOOSE0">in other libraries</a> about graph theory</li> <li><a rel="nofollow" class="external text" href="http://www.martinbroadhurst.com/Graph-algorithms.html">A list of graph algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190713044421/http://www.martinbroadhurst.com/Graph-algorithms.html">Archived</a> 2019-07-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> with references and links to graph library implementations</li></ul> <div class="mw-heading mw-heading3"><h3 id="Online_textbooks">Online textbooks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graph_theory&action=edit&section=32" title="Edit section: Online textbooks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0602129">Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs</a> (2006) by Hartmann and Weigt</li> <li><a rel="nofollow" class="external text" href="http://www.cs.rhul.ac.uk/books/dbook/">Digraphs: Theory Algorithms and Applications</a> 2007 by Jorgen Bang-Jensen and Gregory Gutin</li> <li><a rel="nofollow" class="external text" href="http://diestel-graph-theory.com/index.html">Graph Theory, by Reinhard Diestel</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed reality</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual reality</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" title="Enterprise software">Enterprise software</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational 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