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# Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Jul 2022

Given a graph and two vertex sets satisfying a certain feasibility condition,a reconfiguration problem asks whether we can reach one vertex set from theother by repeating prescribed modification steps while maintaining feasibility.In this setting, Mouawad et al. [IPEC 2014] presented an algorithmicmeta-theorem for reconfiguration problems that says if the feasibility can beexpressed in monadic second-order logic (MSO), then the problem isfixed-parameter tractable parameterized by $\textrm{treewidth} + \ell$, where$\ell$ is the number of steps allowed to reach the target set. On the otherhand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if $\ell$ is notpart of the parameter, then the problem is PSPACE-complete even on graphs ofbounded bandwidth. In this paper, we present the first algorithmic meta-theorems for the casewhere $\ell$ is not part of the parameter, using some structural graphparameters incomparable with bandwidth. We show that if the feasibility isdefined in MSO, then the reconfiguration problem under the so-called tokenjumping rule is fixed-parameter tractable parameterized by neighborhooddiversity. We also show that the problem is fixed-parameter tractableparameterized by $\textrm{treedepth} + k$, where $k$ is the size of sets beingtransformed. We finally complement the positive result for treedepth by showingthat the problem is PSPACE-complete on forests of depth $3$.

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