Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, as reported by Mouawad et al. (IPEC, Springer, Berlin, 2014) presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by \(\text {treewidth} + \ell \), where \(\ell \) is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna (J Comput Syst Sci 93:1–10, 2018). https://doi.org/10.1016/j.jcss.2017.11.003) that if \(\ell \) is not part of the parameter, then the problem is PSPACE-complete even on graphs of constant bandwidth. In this paper, we present the first algorithmic meta-theorems for the case where \(\ell \) is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by \(\text {treedepth} + k\), where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3.