Tag: GiannosStamoulis

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 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 ${\mathsf{s}\mathsf{-}\mathsf{sdp}}_k(x_1,y_1,\dots ,x_k,y_k),$ 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.
Joint work with Petr A. Golovach and Dimitrios M. Thilikos.