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

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate dpk(x1,y1,…, xk,yk), expressing the existence of internally vertex-disjoint paths between Xi and yi, for i ∈ 1,…, 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-sdpk(x1,y1,…,xk,yk), 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. * The full version of the paper can be accessed at https://arxiv.org/abs/2211.01723