Characterizing 2-crossing-critical graphs

TL;DR

This paper fully characterizes 2-crossing-critical graphs, especially focusing on 3-connected cases, their subdivisions, and the finiteness of certain subclasses, advancing understanding of graph crossing numbers.

Contribution

It classifies all 3-connected 2-crossing-critical graphs with specific subdivisions and shows finiteness results for certain subclasses.

Findings

All 3-connected 2-crossing-critical graphs containing a subdivision of V_{10} are characterized.

Methods to derive non 3-connected 2-crossing-critical graphs from 3-connected ones are provided.

Finitely many 3-connected 2-crossing-critical graphs do not contain a subdivision of V_{10}.

Abstract

It is very well-known that there are precisely two minimal non-planar graphs: $K_5$ and $K_{3,3}$ (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the M\"obius Ladder $V_{10}$; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of $V_{10}$; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of $V_{8}$.