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.
@InProceedings{dvorak_et_al:LIPIcs.SoCG.2018.33, author = {Dvor\'{a}k, Zdenek and Hlinen\'{y}, Petr and Mohar, Bojan}, title = {{Structure and Generation of Crossing-Critical Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.33}, URN = {urn:nbn:de:0030-drops-87460}, doi = {10.4230/LIPIcs.SoCG.2018.33}, annote = {Keywords: crossing number, crossing-critical, path-width, exhaustive generation} }
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