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arXiv · 2112.04854

Properties of Large 2-Crossing-Critical Graphs

Drago Bokal·Markus Chimani·Alexander Nover·Jöran Schierbaum·Tobias Stolzmann·Mirko H. Wagner·Tilo Wiedera
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Abstract

A $c$-crossing-critical graph is one that has crossing number at least $c$ but each of its proper subgraphs has crossing number less than $c$. Recently, a set of explicit construction rules was identified by Bokal, Oporowski, Richter, and Salazar to generate all large $2$-crossing-critical graphs (i.e., all apart from a finite set of small sporadic graphs). They share the property of containing a generalized Wagner graph $V_{10}$ as a subdivision. In this paper, we study these graphs and establish their order, simple crossing number, edge cover number, clique number, maximum degree, chromatic number, chromatic index, and treewidth. We also show that the graphs are linear-time recognizable and that all our proofs lead to efficient algorithms for the above measures.

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Drago Bokal, Markus Chimani, Alexander Nover, Jöran Schierbaum, Tobias Stolzmann, Mirko H. Wagner, Tilo Wiedera. 2021-12-09. Properties of Large 2-Crossing-Critical Graphs. https://arxiv.org/abs/2112.04854

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