Automated Testing and Interactive Construction of Unavoidable Sets for Graph Classes of Small Path-width

TL;DR

This paper introduces an interactive framework for finding unavoidable sets in graph classes of small path-width, with a focus on cubic graphs, enabling the determination of extremal girth values and specific extremal graphs.

Contribution

The paper presents a novel interactive framework for constructing unavoidable sets in graph classes, especially cubic graphs, and applies it to determine extremal girth values and graphs for small path-widths.

Findings

Determined all extremal girth values for cubic graphs of path-width 3 to 10.

Identified all smallest graphs with these extremal girth values.

Characterized extremal cubic graphs of path-width 3 and girth 4.

Abstract

We present an interactive framework that, given a membership test for a graph class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width $k$ for all $k \in \{3,\dots, 10\}$. Moreover, we determine all smallest graphs which take on these extremal girth values. As a further application of our framework we characterise the extremal cubic graphs of path-width 3 and girth 4.