Model-Checking for First-Order Logic with Disjoint Paths Predicates in Proper Minor-Closed Graph Classes

Abstract

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate ${\mathsf{dp}}_k(x_1,y_1,\dots ,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in \{1,\dots ,k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every proper minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate $s{-\mathsf{sdp}}_k(x_1,y_1,\dots ,x_k,y_k),$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus.

An extended abstract of this paper appeared in the Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023)