Pathographs and some (un)decidability results

2025-05-26

AI Summary

This paper investigates the realizability problem within the pathograph framework: given a template pathograph $mathfrak{H}$ and a forbidden set $mathcal{F}$ of pathographs, does there exist an $mathcal{F}$-free realization of $mathfrak{H}$? This problem unifies modeling of graph class containment under constraints such as forbidden subgraphs or minors. The authors introduce two novel adjacency types—“spokes” and “rungs”—generalizing s-graphs. They prove the general problem is undecidable, and identify two decidable subclasses: (1) when no rungs are present, admitting a polynomial-time algorithm; and (2) when $mathcal{F}$ is closed under extension, ensuring decidability. Their approach integrates combinatorial graph theory, logical decidability analysis, and counterexample encoding. The results establish a new paradigm for automating proofs of graph decomposition theorems.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesMachine Learning: Graph-based Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsWeb Mining and Content Analysis: Models for Web evolution

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 $Ssubseteq S'$? 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.

Problem

• Introducing pathographs to study graph classes with forbidden structures
• Deciding if a pathograph has an F-free realization is undecidable generally
• Pathograph realization is decidable for no-rung cases or edge-spoke-rung closed F

Innovation

• Introduces pathographs with extra adjacency relations
• Studies decidability of pathograph realization problem
• Decidable cases include no rungs or closed sets