Abstract
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 \(\textit{Q}\) asks whether it is possible to transform one feasible solution for \(\textit{Q}\) into another in at most \(\ell \) steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for \(\textit{Q}\) of size bounded by \(k\). Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by \(\ell \), 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 \(\ell + t\) 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.
A.E. Mouawad and N. Nishimura—Research supported by the Natural Science and Engineering Research Council of Canada.
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Mouawad, A.E., Nishimura, N., Raman, V., Wrochna, M. (2014). Reconfiguration over Tree Decompositions. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_21
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