Graph Minors and Minimum Degree

TL;DR

This paper explores the structure and properties of graph classes defined by minor-minimal forbidden graphs related to minimum degree, providing bounds, characterizations, and constructions for various graph parameters.

Contribution

It characterizes the forbidden minors for classes of graphs with bounded minimum degree, including bounds, constructions, and relationships with connectivity and other parameters.

Findings

Every (k+1)-regular graph with fewer than (4/3)(k+2) vertices is in the forbidden minor set.

Characterization of forbidden minors obtained by adding one vertex to smaller forbidden minors.

Existence of forbidden minors with low connectivity and arbitrary block structures.

Abstract

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$. This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex. For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.