Large Non-Planar Graphs and an Application to Crossing-Critical Graphs

TL;DR

This paper establishes structural properties of large non-planar graphs, showing they contain specific minors or subdivisions, and applies these results to classify certain crossing-critical graphs.

Contribution

It proves that large 4-connected non-planar graphs necessarily contain specific minors or subdivisions, advancing understanding of graph minors and crossing-critical graphs.

Findings

Large non-planar graphs contain specific minors or subdivisions.

Finiteness of certain 3-connected 2-crossing-critical graphs is established.

Structural characterization aids in understanding crossing-critical graph classes.

Abstract

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.