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arXiv · 1803.01931

Structure and generation of crossing-critical graphs

Zdeněk Dvořák·Petr Hliněný·Bojan Mohar
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Abstract

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.

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Zdeněk Dvořák, Petr Hliněný, Bojan Mohar. 2018-03-05. Structure and generation of crossing-critical graphs. https://arxiv.org/abs/1803.01931

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