A Logic for Minor-Free Graph Classes: Model Checking, Dependence, and Combinatorial Reconfiguration
Year: 2026
Type: preprint
Abstract: We introduce \emph{sub-connectivity logic}, denoted by $\textsf{FOscon}$, an extension of first-order logic for graphs with a specified set $Z$ of admissible edges. Its additional atom $\textsf{scon}(x,y;\bar z)$ asserts that $x$ and $y$ are connected by a path using only edges of $Z$ and avoiding the vertices in $\bar z$. Our main structural result states that, for every weakly sparse graph class $\mathscr C$, the class of all expansions of graphs in $\mathscr C$ by rooted spanning forest orders has bounded twin-width if and only if $\mathscr C$ excludes a fixed minor. When $\mathscr C$ excludes a fixed minor, we can also compute an additional linear order that preserves bounded twin-width in polynomial time. We translate $\textsf{FOscon}$ into first-order logic over an expansion by a depth-first spanning forest order of the admissible-edge graph. Combining this translation with our structural result, we obtain a model checking algorithm for $\textsf{FOscon}$ with running time $f(|ϕ|,h(G))\cdot |G|^c$, where $f$ is computable, $c$ is an absolute constant, and $h(G)$ is the Hadwiger number of $G$. The translation and model-checking algorithm extend to binary relational structures, with the Hadwiger number measured on the Gaifman graph. For a graph class $\mathscr C$ let $\mathscr C_Z:=\{(G,Z):G\in\mathscr C,\ Z\subseteq E(G)\}$. We show that for every weakly sparse graph class $\mathscr C$, the class $\mathscr C_Z$ is monadically dependent for $\textsf{FOscon}$ if and only if $\mathscr C$ excludes a fixed minor. For monotone classes that admit efficient minor encodings, a corresponding hardness result makes this frontier computationally tight. We give applications to combinatorial reconfiguration and solution discovery problems, and study a guarded extension of $\textsf{FOscon}$ motivated by database queries.
Source: arXiv (Cornell University)
Institutions: University of Warsaw, University of Bremen, Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Recherche en Informatique Fondamentale, Université Clermont Auvergne, Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes, Institut national du patrimoine
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220650046