Properties of Large 2-Crossing-Critical Graphs

TL;DR

This paper thoroughly analyzes large 2-crossing-critical graphs, establishing their key properties and demonstrating that they can be efficiently recognized and characterized through algorithms.

Contribution

It provides a comprehensive study of large 2-crossing-critical graphs, including their structural properties and efficient recognition algorithms.

Findings

Graphs contain a subdivision of generalized Wagner graph V_{10}

Properties such as crossing number, clique number, and treewidth are characterized

Recognition of these graphs can be performed in linear time

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.