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Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div><div class="paragraphbreak" style="margin-top:0.5em"></div> <p>This block affects editing on all Wikimedia wikis. </p><p>The IP address or range 8.222.128.0/17 has been globally <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:Jon_Kolbert" title="User:Jon Kolbert">‪Jon Kolbert‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <p><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/NOP" class="extiw" title="m:Special:MyLanguage/NOP">Open proxy/Webhost</a>: See the <a href="https://meta.wikimedia.org/wiki/WM:OP/H" class="extiw" title="m:WM:OP/H">help page</a> if you are affected </p> </div> <p>This block will expire on 15:12, 27 August 2028. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even while globally blocked, you will <i>usually</i> still be able to edit pages on <a href="https://meta.wikimedia.org/wiki/" class="extiw" title="m:">Meta-Wiki</a>. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>If you believe you were blocked by mistake, you can find additional information and instructions in the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards/Wizard" class="extiw" title="m:Special:MyLanguage/Stewards/Wizard">Stewards Block Wizard</a>. <a href="/w/index.php?title=MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward&action=edit&redlink=1" class="new" title="MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward (page does not exist)">MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward</a> </p> </div> <p>Other useful links: <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">Global blocks</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li></ul><hr /> <div id="viewsourcetext">You can view and copy the source of this page:</div><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="en" dir="ltr" name="wpTextbox1">{{Short description|Bijection between the vertex set of two graphs}} In [[graph theory]], an '''isomorphism of [[Graph (discrete mathematics)|graphs]]''' ''G'' and ''H'' is a [[bijection]] between the vertex sets of ''G'' and ''H'' :<math> f \colon V(G) \to V(H) </math> such that any two vertices ''u'' and ''v'' of ''G'' are [[Adjacent (graph theory)|adjacent]] in ''G'' [[if and only if]] <math>f(u)</math> and <math>f(v)</math> are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of [[isomorphism]] being a structure-preserving bijection. If an [[isomorphism]] exists between two graphs, then the graphs are called '''isomorphic''' and denoted as <math>G\simeq H</math>. In the case when the isomorphism is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the isomorphism is called an [[graph automorphism|automorphism]] of ''G''. Graph isomorphism is an [[equivalence relation]] on graphs and as such it partitions the [[class (set theory)|class]] of all graphs into [[equivalence class]]es. A set of graphs isomorphic to each other is called an '''[[isomorphism class]]''' of graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in computer science, known as the [[graph isomorphism problem]].<ref>{{Cite journal |last=Grohe |first=Martin |date=2020-11-01 |title=The Graph Isomorphism Problem |volume=63 |pages=128–134 |journal=[[Communications of the ACM]] |issue=11 |url=https://cacm.acm.org/magazines/2020/11/248220-the-graph-isomorphism-problem/abstract |access-date=2023-03-06 |doi=10.1145/3372123}}</ref><ref>{{Cite news |last=Klarreich |first=Erica |date=2015-12-14 |title=Landmark Algorithm Breaks 30-Year Impasse |work=[[Quanta Magazine]] |url=https://www.quantamagazine.org/algorithm-solves-graph-isomorphism-in-record-time-20151214/ |access-date=2023-03-06}}</ref> The two graphs shown below are isomorphic, despite their different looking [[graph drawing|drawings]]. {|class="wikitable" style="margin: 1em auto 1em auto" ! Graph G ! Graph H ! An isomorphism<br />between G and H |- |style="padding-left:2em;padding-right:2em;"|[[File:Graph isomorphism a.svg|class=skin-invert|100px]] |style="padding-left:1em;padding-right:1em;"|[[File:Graph isomorphism b.svg|class=skin-invert|210px]] |align="center" |''f''(''a'') = 1 ''f''(''b'') = 6 ''f''(''c'') = 8 ''f''(''d'') = 3 ''f''(''g'') = 5 ''f''(''h'') = 2 ''f''(''i'') = 4 ''f''(''j'') = 7 |} ==Variations== In the above definition, graphs are understood to be [[directed graph|undirected]] [[labeled graph|non-labeled]] [[weighted graph|non-weighted]] graphs. However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception. ===Isomorphism of labeled graphs=== For [[labeled graph]]s, two definitions of isomorphism are in use. Under one definition, an isomorphism is a vertex bijection which is both edge-preserving and label-preserving.<ref>[https://books.google.com/books?id=14138OJXzy4C&pg=PA424 p.424]</ref><ref>{{cite book | chapter-url=https://link.springer.com/chapter/10.1007/11751649_46 | doi=10.1007/11751649_46 | chapter=Efficient Method to Perform Isomorphism Testing of Labeled Graphs | title=Computational Science and Its Applications - ICCSA 2006 | series=Lecture Notes in Computer Science | date=2006 | last1=Hsieh | first1=Shu-Ming | last2=Hsu | first2=Chiun-Chieh | last3=Hsu | first3=Li-Fu | volume=3984 | pages=422–431 | isbn=978-3-540-34079-9 }}</ref> Under another definition, an isomorphism is an edge-preserving vertex bijection which preserves equivalence classes of labels, i.e., vertices with equivalent (e.g., the same) labels are mapped onto the vertices with equivalent labels and vice versa; same with edge labels.<ref>Pierre-Antoine Champin, Christine Solnon, [https://link.springer.com/chapter/10.1007/3-540-45006-8_9 "Measuring the Similarity of Labeled Graphs"] in: ''[[Lecture Notes in Computer Science]]'', vol. 2689, pp 80–95</ref> For example, the <math>K_2</math> graph with the two vertices labelled with 1 and 2 has a single automorphism under the first definition, but under the second definition there are two auto-morphisms. The second definition is assumed in certain situations when graphs are endowed with ''unique labels'' commonly taken from the integer range 1,...,''n'', where ''n'' is the number of the vertices of the graph, used only to uniquely identify the vertices. In such cases two labeled graphs are sometimes said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic (otherwise the definition of isomorphism would be trivial). ==Motivation== The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used: [[digraph (mathematics)|digraph]]s, [[labeled graph]]s, [[colored graph]]s, [[rooted tree]]s and so on. The isomorphism relation may also be defined for all these generalizations of graphs: the isomorphism bijection must preserve the elements of structure which define the object type in question: [[arc (graph theory)|arc]]s, labels, vertex/edge colors, the root of the rooted tree, etc. The notion of "graph isomorphism" allows us to distinguish [[graph properties]] inherent to the structures of graphs themselves from properties associated with graph representations: [[graph drawing]]s, [[graph (data structure)|data structures for graphs]], [[graph labeling]]s, etc. For example, if a graph has exactly one [[cycle (graph theory)|cycle]], then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (''represented'' by) the [[integer]]s 1, 2,... ''N'', then the expression :<math>\sum_{v \in V(G)} v\cdot\text{deg }v</math> may be different for two isomorphic graphs. == Whitney theorem == {{main article|Whitney graph isomorphism theorem}} [[File:Whitneys theorem exception.svg|right|thumb|200px|The exception to Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs.]] The '''Whitney graph isomorphism theorem''',<ref>{{cite journal|last=Whitney|first=Hassler|title=Congruent Graphs and the Connectivity of Graphs|journal=American Journal of Mathematics|date=January 1932|volume=54|issue=1|pages=150–168|jstor=2371086|doi=10.2307/2371086|hdl=10338.dmlcz/101067|hdl-access=free}}</ref> shown by [[Hassler Whitney]], states that two connected graphs are isomorphic if and only if their [[line graph]]s are isomorphic, with a single exception: ''K''<sub>3</sub>, the [[complete graph]] on three vertices, and the [[complete bipartite graph]] ''K''<sub>1,3</sub>, which are not isomorphic but both have ''K''<sub>3</sub> as their line graph. The Whitney graph theorem can be extended to [[hypergraph]]s.<ref>Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs. J. Comb. Theory, Ser. B 71(2): 215–230. 1997.</ref> ==Recognition of graph isomorphism== {{main article|Graph isomorphism problem}} While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem. Its practical applications include primarily [[cheminformatics]], [[mathematical chemistry]] (identification of chemical compounds), and [[electronic design automation]] (verification of equivalence of various representations of the design of an [[electronic circuit]]). The graph isomorphism problem is one of few standard problems in [[computational complexity theory]] belonging to [[NP (complexity)|NP]], but not known to belong to either of its well-known (and, if [[P versus NP problem|P&nbsp;≠&nbsp;NP]], disjoint) subsets: [[P (complexity)|P]] and [[NP-complete]]. It is one of only two, out of 12 total, problems listed in {{harvtxt|Garey|Johnson|1979}} whose complexity remains unresolved, the other being [[integer factorization]]. It is however known that if the problem is NP-complete then the [[polynomial hierarchy]] collapses to a finite level.<ref>{{cite journal | title=Graph isomorphism is in the low hierarchy | first=Uwe | last=Schöning | journal=[[Journal of Computer and System Sciences]] | volume=37 | issue=3 | year=1988 | pages=312–323 | doi=10.1016/0022-0000(88)90010-4| doi-access= }}</ref> In November 2015, [[László Babai]], a mathematician and computer scientist at the University of Chicago, claimed to have proven that the graph isomorphism problem is solvable in [[quasi-polynomial time]].<ref>{{citation|doi=10.1126/science.aad7416|journal=[[Science (journal)|Science]]|first=Adrian|last=Cho|date=November 10, 2015|title=Mathematician claims breakthrough in complexity theory}}.</ref><ref>{{citation|last=Klarreich|first=Erica|url=https://www.quantamagazine.org/20151214-graph-isomorphism-algorithm/|title=Landmark Algorithm Breaks 30-Year Impasse|journal=[[Quanta Magazine]]|date=December 14, 2015}}</ref> He published preliminary versions of these results in the proceedings of the 2016 [[Symposium on Theory of Computing]],<ref>{{citation|last=Babai|first=László|contribution=Graph isomorphism in quasipolynomial time [extended abstract]|doi=10.1145/2897518.2897542|mr=3536606|pages=684–697|publisher=ACM, New York|title=STOC'16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing|year=2016|isbn=978-1-4503-4132-5 |s2cid=17118954 }}</ref> and of the 2018 [[International Congress of Mathematicians]].<ref>{{citation|last=Babai|first=László|contribution=Group, graphs, algorithms: the graph isomorphism problem|mr=3966534|pages=3319–3336|publisher=World Sci. Publ., Hackensack, NJ|title=Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures|year=2018}}</ref> In January 2017, Babai briefly retracted the quasi-polynomiality claim and stated a [[sub-exponential time]] complexity bound instead. He restored the original claim five days later.<ref>{{citation|last=Babai|first= László|url=http://people.cs.uchicago.edu/~laci/update.html|title=Graph isomorphism update|date=January 9, 2017}}</ref> {{as of|2024}}, the full journal version of Babai's paper has not yet been published. Its generalization, the [[subgraph isomorphism problem]], is known to be NP-complete. The main areas of research for the problem are design of fast algorithms and theoretical investigations of its [[Analysis of algorithms|computational complexity]], both for the general problem and for special classes of graphs. The [[Weisfeiler Leman graph isomorphism test]] can be used to heuristically test for graph isomorphism.<ref>{{cite book|arxiv=2201.07083 |doi=10.1109/ICASSP39728.2021.9413523 |chapter=A Short Tutorial on the Weisfeiler-Lehman Test and Its Variants |title=ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) |date=2021 |last1=Huang |first1=Ningyuan Teresa |last2=Villar |first2=Soledad |pages=8533–8537 |isbn=978-1-7281-7605-5 |s2cid=235780517 }}</ref> If the test fails the two input graphs are guaranteed to be non-isomorphic. If the test succeeds the graphs may or may not be isomorphic. There are generalizations of the test algorithm that are guaranteed to detect isomorphisms, however their run time is exponential. Another well-known algorithm for graph isomorphism is the vf2 algorithm, developed by Cordella et al. in 2001.<ref>{{cite journal|last1=Cordella|first1=L. P.|last2=Foggia|first2=P.|last3=Sansone|first3=C.|last4=Vento|first4=M.|title=An Improved Algorithm for Matching Large Graphs|journal=3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition|pages=149–159|year=2001|url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=f3e10bd7521ec6263a58fdaa4369dfe8ad50888c}}</ref> The vf2 algorithm is a depth-first search algorithm that tries to build an isomorphism between two graphs incrementally. It uses a set of feasibility rules to prune the search space, allowing it to efficiently handle graphs with thousands of nodes. The vf2 algorithm has been widely used in various applications, such as pattern recognition, computer vision, and bioinformatics. While it has a worst-case exponential time complexity, it performs well in practice for many types of graphs. ==See also== * [[Graph homomorphism]] * [[Graph automorphism problem]] *[[Graph isomorphism problem]] * [[Graph canonization]] * [[Fractional graph isomorphism]] ==Notes== {{reflist|29em}} ==References== * {{Garey-Johnson}} {{DEFAULTSORT:Graph Isomorphism}} [[Category:Graph theory]] [[Category:Graph algorithms]] [[Category:Morphisms]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:As_of" title="Template:As of">Template:As of</a> (<a href="/w/index.php?title=Template:As_of&action=edit" title="Template:As of">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Citation" 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class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>Add languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> 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