Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs

We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topological subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input formula and the width of the poset. Results of this form, i.e. showing that model-checking for a certain logic is tractable on a certain class of structures, are often referred to as algorithmic meta-theorems since they give a unified proof for the tractability of a whole range of problems. First-order logic is arguably one of the most important logics in this context since it is powerful enough to express many computational problems (e.g. the existence of cliques, dominating sets etc.) and yet its model-checking problem is tractable on rich classes of graphs. In fact, Grohe et al have shown that model-checking for FO is tractable on all nowhere dense classes of graphs. Successor-invariant FO is a semantic extension of FO by allowing the use of an additional binary relation which is interpreted as a directed Hamiltonian cycle, restricted to formulae whose truth value does not depend on the specific choice of a Hamiltonian cycle. While this is very natural in the context of model-checking (after all, storing a structure in computer memory usually brings with it a linear order on the structure), the question of how the computational complexity of the model-checking problem for this richer logic compares to that of plain FO is still open. Our result for successor-invariant FO extends previous results for this logic on planar graphs and graphs with excluded minors, further narrowing the gap between what is known for FO and what is known for successor-invariant FO. The proof uses Grohe and Marx's structure theorem for graphs with excluded topological subgraphs. For order-invariant FO we show that Gajarský et al.'s recent result for FO carries over to order-invariant FO.