Properties of Large 2-Crossing-Critical Graphs
Drago Bokal, Markus Chimani, Alexander Nover, J├Âran Schierbaum, Tobias Stolzmann, Mirko H. Wagner, and Tilo Wiedera
Vol. 26, no. 1, pp. 111-147, 2022. Regular paper.
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.

Keywords. Crossing number, crossing-critical graph, chromatic number, chromatic index, treewidth.

 This work is licensed under the terms of the CC-BY license.
Submitted: May 2020.
Reviewed: December 2020.
Revised: February 2021.
Reviewed: June 2021.
Revised: July 2021.
Accepted: March 2022.
Final: March 2022.
Published: April 2022.
Communicated by Alexander Wolff
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