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

TL;DR

This paper proves that model-checking for an extended disjoint paths logic in proper minor-closed graph classes and graphs with bounded Euler genus can be performed efficiently in quadratic time.

Contribution

It introduces and analyzes the complexity of FOL+DP and FOL+SDP logics, establishing quadratic-time model-checking algorithms for these logics in specific graph classes.

Findings

Model-checking for FOL+DP is quadratic-time in proper minor-closed classes.

Model-checking for FOL+SDP is quadratic-time in graphs with bounded Euler genus.

The techniques extend the applicability of logic-based graph problem solving.

Abstract

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in\{1,\ldots, k\}$. 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 $s{\sf -sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ 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.