Model Checking Disjoint-Paths Logic on Topological-Minor-Free Graph Classes
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Disjoint-paths logic, denoted 𝖥𝖮+𝖣𝖯, extends first-order logic (𝖥𝖮) with atomic predicates 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 1≤ i≤ k. We prove that for every graph class excluding some fixed graph as a topological minor, the model checking problem for 𝖥𝖮+𝖣𝖯 is fixed-parameter tractable. This essentially settles the question of tractable model checking for this logic on subgraph-closed classes, since the problem is hard on subgraph-closed classes not excluding a topological minor (assuming a further mild condition of efficiency of encoding).
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