Pathographs and some (un)decidability results
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915305505161216 |
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| author | Carter, Daniel Trotignon, Nicolas |
| author_facet | Carter, Daniel Trotignon, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pathographs and some (un)decidability results Carter, Daniel Trotignon, Nicolas Combinatorics Discrete Mathematics Data Structures and Algorithms 05C85 (Primary) 68Q45, 05C75 (Secondary) 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. |
| title | Pathographs and some (un)decidability results |
| topic | Combinatorics Discrete Mathematics Data Structures and Algorithms 05C85 (Primary) 68Q45, 05C75 (Secondary) |
| url | https://arxiv.org/abs/2505.19871 |