Reconfiguration over Tree Decompositions

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem Q asks whether it is possible to transform one feasible solution for Q into another in at most l steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for Q of size bounded by k. Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by l, we investigate the complexity of such problems restricted to graphs of bounded treewidth. We show that the reconfiguration versions of most vertex-subset problems remain PSPACE-complete on graphs of treewidth at most t but are fixed-parameter tractable parameterized by l+ t for all vertex-subset problems definable in monadic second-order logic (MSOL). To prove the latter result, we introduce a technique which allows us to circumvent cardinality constraints and define reconfiguration problems in MSOL.