On December 6, 2022, Giannos Stamoulis from Université de Montpellier gave at the Discrete Math Seminar on a fixed-parameter tractable algorithm for the model checking for the first-order logic extended with disjoint paths predicates in H-minor-free graphs. The title of his talk was “Model-Checking for First-Order Logic with Disjoint Paths Predicates in Proper Minor-Closed Graph Classes“.

## Giannos Stamoulis, 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 $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ 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 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 $\mathsf{s-sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ 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.

Joint work with Petr A. Golovach and Dimitrios M. Thilikos.