Contractible edges in 3-connected graphs that preserve a minor

TL;DR

This paper proves that in certain 3-connected graphs with a minor, there exists a large forest of edges that can be contracted while preserving 3-connectivity and the minor, generalizing previous results and including a splitter theorem.

Contribution

It introduces a new theorem on graph minors and contractible edges, extending prior work by providing sharper bounds and conditions.

Findings

Existence of large forests of contractible edges in 3-connected graphs with minors

Improved bounds for contractible edges in triangle-free graphs

Establishment of a splitter theorem for graph minors

Abstract

Let $G$ be a $3$-connected graph with a $3$-connected (or sufficiently small) simple minor $H$. We establish that $G$ has a forest $F$ with at least $\left\lceil(|G|-|H|+1)/2\right\rceil$ edges such that $G/e$ is $3$-connected with an $H$-minor for each $e\in E(F)$. Moreover, we may pick $F$ with $|G|-|H|$ edges provided $G$ is triangle-free. These results are sharp. Our result generalizes a previous one by Ando et. al., which establishes that a $3$-connected graph $G$ has at least $\left\lceil|G|/2\right\rceil$ contractible edges. As another consequence, each triangle-free $3$-connected graph has an spanning tree of contractible edges. Our results follow from a more general theorem on graph minors, a splitter theorem, which is also established here.