On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids

TL;DR

This paper investigates specific minors related to graph planarity, proving that certain 3-connected graphs contain minors with particular properties, and extends these results to regular matroids, enhancing understanding of graph and matroid obstructions.

Contribution

It establishes new minor existence results for 3-connected graphs and generalizes these findings to regular matroids, advancing the theory of graph and matroid obstructions.

Findings

Existence of $K_5$-minors containing a given triangle in 3-connected graphs.

Identification of minors related to $K_{3,3}$ with added edges in non-planar graphs.

Extension of these minor results to the class of regular matroids.

Abstract

In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is that if $G$ is a 3-connected simple non-planar graph not isomorphic to $K_5$ and $e,f\in E(G)$, then $G$ has a minor $H$ such that $e,f\in E(H)$ and, up to isomorphisms, $H$ is one of the four non-isomorphic simple graphs obtained from $K_{3,3}$ by the addiction of \,0, 1 or 2 edges. We generalize this second result to the class of the regular matroids.