Model Checking for Low Monodimensionality Fragments of CMSO on Topological-Minor-Free Graph Classes

2026-04-30

AI Summary

This work addresses the challenge of efficiently evaluating rich fragments of Counting Monadic Second-Order logic (CMSO) on topological minor-free graph classes, where existing methods fall short. We introduce a novel CMSO fragment that restricts set quantifiers to vertex sets of bounded monotone dimension and incorporates a disjoint-paths predicate. By leveraging a framework of annotated graph parameters together with structural graph theory, we extend such quantifier restrictions—previously limited to first-order logic—to topological minor-free classes for the first time. Our results establish fixed-parameter tractability of model checking for this fragment on these graph classes, thereby generalizing several classical meta-theorems beyond the expressive limits of first-order logic.

Abstract

Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model checking on graph classes excluding a fixed topological minor, comparable results for richer fragments of CMSO were previously unknown. We further develop the framework of Sau, Stamoulis, and Thilikos [SODA 2025] for fragmenting CMSO via annotated graph parameters, which restrict set quantification to vertex sets satisfying bounded structural conditions. Following this approach, we identify a fragment of CMSO, namely the one defined by allowing quantification only over sets having what we call low monodimensionality, that generalizes several previously-known logics and we show that model checking for this fragment, enhanced with the disjoint-paths predicate, is fixed-parameter tractable on topological-minor-free graph classes. Such classes essentially delimit the tractability for this logic on subgraph-closed classes. As a consequence, our results lift several known algorithmic meta-theorems beyond first-order logic to the topological-minor-free setting.

Problem

• model checking
• CMSO
• topological-minor-free
• algorithmic meta-theorems
• low monodimensionality

Innovation

• low monodimensionality
• CMSO fragments
• topological-minor-free graphs
• model checking
• algorithmic meta-theorems