Pathographs and some (un)decidability results

TL;DR

The paper introduces pathographs as a unifying framework to study graph classes defined by forbidding structures across subgraphs, minors, topological minors, and Truemper configurations, and formalizes the pathograph realization problem: given $\mathfrak{H}$ and a finite $\mathcal{F}$, does an $\mathcal{F}$-free realization exist? It proves undecidability in general via a Wang tiling reduction, while establishing decidability in restricted cases such as rungless pathographs and when $\mathcal{F}$ is closed under adding adjacencies; it also provides two proofs (Courcelle-based and explicit automaton) for the rungless case and demonstrates potential applications to decomposition theorems. The results delineate a boundary between decidable and undecidable regimes and offer algorithmic tools for structured graph containment questions, with implications for decomposition theory in graph classes. Overall, the work broadens the toolbox for algorithmic graph theory by embedding diverse containment notions within a single realizability framework and identifying robust decidable subcases guided by structural restrictions.

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 Lévêque--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.

Figures (13)

• Figure 1: An example pathograph $\mathfrak{G}$ (left) and realization $G$ (right). Here, $\mathfrak{G}$ has 5 vertices, 2 urpaths, 2 edges, 1 spoke, and 1 rung.
• Figure 2: Pathographs corresponding to Truemper configurations.
• Figure 3: The directed multicolored pathograph $\mathfrak{H}$ in the case $n=3$. The triple arrow indicates that every possible (directed) edge, spoke, and rung is drawn from the left to the right. There will be a total of 9 edges, 3 spokes, and 1 rung between the sides. The color of the directed edge $x_iy_j$ is $f'(i,j)\in S$.
• Figure 4: An example $\mathcal{F}$. Here, $S$ consists of the two Wang tiles $s,t$ shown at the top of the figure, using colors $C=\{\text{green}, \text{magenta}\}$. Labels of black edges indicate their color.
• Figure 5: Example translation from a directed multicolored pathograph to a vertex-colored pathograph. In this example, $K=3$. The labels of black edges/vertices indicate their color, when applicable. In the rightmost picture, we have just drawn the non-edges between the two sides, represented as dotted lines.

Theorems & Definitions (40)

• theorem 1.1
• theorem 1.2
• corollary 1
• theorem 1.3
• theorem 1.4
• corollary 2
• theorem 1.5
• corollary 3
• proposition 1
• theorem 2.1