Document Zbl 1479.05071

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Geom.</a> <a href="/?q=in%3A473350" title="Articles in this Issue">100, Article ID 101818, 11 p. (2022)</a>. </div> <div class="abstract">Summary: Let $P = ( p_1, \dots, p_n)$ be a geometric path in the plane. For a real number $0 < \gamma < 180 ° $, $P$ is called an angle-monotone path with width $\gamma $ if there exists a closed wedge of angle $\gamma$ such that the vector of every edge $( p_i, p_{i + 1})$ of $P$ lies in this wedge. Let $G$ be a geometric graph in the plane. The graph $G$ is called angle-monotone with width $\gamma $ if there exists an angle-monotone path with width $\gamma$ between any two vertices. Let $\gamma_{\min}$ be the smallest width such that every set of points in the plane has a plane angle-monotone graph with width $\gamma_{\min}$. It has been shown that $90 ° < \gamma_{\min} \leq 120 ° $. In this paper, we show that $\gamma_{\min} = 120 ° $. This solves an open problem posed by <span class="zbmathjax-textit">N. Bonichon</span> et al. [Lect. Notes Comput. Sci. 9801, 519–531 (2016; <a href="/1471.68289">Zbl 1471.68289</a>)].</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C10" title="MSC2020">05C10</a> </td> <td class="space"> Planar graphs; geometric and topological aspects of graph theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68U05" title="MSC2020">68U05</a> </td> <td class="space"> Computer graphics; computational geometry (digital and algorithmic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aangle-monotone+path">angle-monotone path</a>; <a href="/?q=ut%3Aplane+graph">plane graph</a>; <a href="/?q=ut%3Astretch+factor">stretch factor</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1471.68289">Zbl 1471.68289</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1479.05071" data-ciurl="/ci/07445253" data-biburl="/bibtex/07445253.bib" data-amsurl="/amsrefs/07445253.bib" data-xmlurl="/xml/07445253.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07445253.pdf" title="Zbl 1479.05071 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.comgeo.2021.101818" aria-label="DOI for “On the plane angle-monotone graphs”" title="10.1016/j.comgeo.2021.101818">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alamdari, S.; Chan, T. M.; Grant, E.; Lubiw, A.; Pathak, V., Self-approaching graphs, (Didimo, W.; Patrignani, M., Graph Drawing (2013), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 260-271 · <a href="/1377.68158" class="nowrap">Zbl 1377.68158</a></td> </tr><tr> <td>[2]</td> <td class="space">Bakhshesh, D.; Farshi, M., (Weakly) Self-approaching geometric graphs and spanners, Comput. Geom., 78, 20-36 (2019) · <a href="/1457.05069" class="nowrap">Zbl 1457.05069</a></td> </tr><tr> <td>[3]</td> <td class="space">Bakhshesh, D.; Farshi, M., Angle-monotonicity of Delaunay triangulation, Comput. Geom., 94, Article 101711 pp. (2021) · <a href="/1470.68233" class="nowrap">Zbl 1470.68233</a></td> </tr><tr> <td>[4]</td> <td class="space">Bonichon, N.; Bose, P.; Carmi, P.; Kostitsyna, I.; Lubiw, A.; Verdonschot, S., Gabriel triangulations and angle-monotone graphs: local routing and recognition, (Proceedings of the 24th International Symposium on Graph Drawing (GD 2016) (2016)), 519-531 · <a href="/1471.68289" class="nowrap">Zbl 1471.68289</a></td> </tr><tr> <td>[5]</td> <td class="space">Da Lozzo, G.; D’Angelo, A.; Frati, F., On planar greedy drawings of 3-connected planar graphs, (Aronov, B.; Katz, M. 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Appl., 26, 02, 89-110 (2016) · <a href="/1353.68281" class="nowrap">Zbl 1353.68281</a></td> </tr><tr> <td>[8]</td> <td class="space">He, X.; Zhang, H., On succinct convex greedy drawing of 3-connected plane graphs, (Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (2011), SIAM), 1477-1486 · <a href="/1374.68352" class="nowrap">Zbl 1374.68352</a></td> </tr><tr> <td>[9]</td> <td class="space">Icking, C.; Klein, R.; Langetepe, E., Self-approaching curves, (Mathematical Proceedings of the Cambridge Philosophical Society, vol. 125 (1999), Cambridge Univ Press), 441-453 · <a href="/0923.51017" class="nowrap">Zbl 0923.51017</a></td> </tr><tr> <td>[10]</td> <td class="space">Leighton, T.; Moitra, A., Some results on greedy embeddings in metric spaces, Discrete Comput. 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Graph Algorithms Appl., 23, 2, 345-369 (2019) · <a href="/1416.05271" class="nowrap">Zbl 1416.05271</a></td> </tr><tr> <td>[12]</td> <td class="space">Narasimhan, G.; Smid, M., Geometric Spanner Networks (2007), Cambridge University Press · <a href="/1140.68068" class="nowrap">Zbl 1140.68068</a></td> </tr><tr> <td>[13]</td> <td class="space">Nöllenburg, M.; Prutkin, R., Euclidean greedy drawings of trees, (Annual European Symposium on Algorithms (2013), Springer), 767-778 · <a href="/1395.68216" class="nowrap">Zbl 1395.68216</a></td> </tr><tr> <td>[14]</td> <td class="space">Rao, A.; Ratnasamy, S.; Papadimitriou, C.; Shenker, S.; Stoica, I., Geographic routing without location information, (Proceedings of the 9th Annual International Conference on Mobile Computing and Networking. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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