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