Pathographs and some (un)decidability results

TL;DR

This paper introduces pathographs, a new framework for studying graph classes defined by forbidden structures, and explores the decidability of related realization problems, showing undecidability in general but decidability under certain restrictions.

Contribution

The paper defines pathographs as a generalization of s-graphs, introduces a decision problem for their realizations, and establishes conditions under which this problem is decidable or undecidable.

Findings

Pathograph realization problem is undecidable in general.

Decidable when pathographs have no rungs or when the set of forbidden pathographs is closed under certain operations.

Potential applications to graph decomposition theorems.

Abstract

We introduce pathographs as a framework to study graph classes defined by forbidden structures, including forbidding induced subgraphs, minors, etc. Pathographs approximately generalize s-graphs of L\'ev\^eque--Lin--Maffray--Trotignon by the addition of two extra adjacency relations: one between subdivisible edges and vertices called spokes, and one between pairs of subdivisible edges called rungs. We consider the following decision problem: given a pathograph $\mathfrak{H}$ and a finite set of pathographs $\mathcal{F}$, is there an $\mathcal{F}$-free realization of $\mathfrak{H}$? This may be regarded as a generalization of the "graph class containment problem": given two graph classes $S$ and $S'$, is it the case that $S\subseteq S'$? We prove the pathograph realization problem is undecidable in general, but it is decidable in the case that $\mathfrak{H}$ has no rungs (but may have spokes), or if $\mathcal{F}$ is closed under adding edges, spokes, and rungs. We also discuss some potential applications to proving decomposition theorems.