Reconfiguration over tree decompositions
Amer E. Mouawad, Naomi Nishimura, Venkatesh Raman, Marcin, Wrochna

TL;DR
This paper studies the complexity of reconfiguring feasible solutions in vertex-subset problems on graphs with bounded treewidth, showing PSPACE-completeness generally but fixed-parameter tractability for MSOL-definable problems.
Contribution
It proves PSPACE-completeness of reconfiguration problems on bounded treewidth graphs and introduces a technique to handle MSOL-definable problems for fixed-parameter tractability.
Findings
Reconfiguration problems are PSPACE-complete on graphs of bounded treewidth.
MSOL-definable problems are fixed-parameter tractable when parameterized by solution steps and treewidth.
A new technique allows MSOL-based reconfiguration problem formulation despite cardinality constraints.
Abstract
A vertex-subset graph problem defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem asks whether it is possible to transform one feasible solution for into another in at most steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for of size bounded by . Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by , 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 but are fixed-parameter tractable parameterized by for all vertex-subset problems definable in monadic…
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