2026-07-22T03:56:50Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8746 2024-03-06T10:42:33Z ddc:004 open_access

Structure and Generation of Crossing-Critical Graphs Dvorák, Zdenek Hlinený, Petr Mohar, Bojan crossing number crossing-critical path-width exhaustive generation We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Zdenek Dvorák and Petr Hlinený and Bojan Mohar 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.33 urn:nbn:de:0030-drops-87460 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.33 eng https://creativecommons.org/licenses/by/3.0/legalcode