On finding short reconfiguration sequences between independent sets
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Assume we are given a graph G, two independent sets S and T in G of size k ≥ 1, and a positive integer ℓ≥ 1. The goal is to decide whether there exists a sequence ⟨ I_0, I_1, ..., I_ℓ⟩ of independent sets such that for all j ∈{0,…,ℓ-1} the set I_j is an independent set of size k, I_0 = S, I_ℓ = T, and I_j+1 is obtained from I_j by a predetermined reconfiguration rule. We consider two reconfiguration rules. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most ℓ steps that transforms S into T, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex. In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph. Both TSO and TJO are known to be fixed-parameter tractable when parameterized by ℓ on nowhere dense classes of graphs. In this work, we show that both problems are fixed-parameter tractable for parameter k + ℓ + d on d-degenerate graphs as well as for parameter |M| + ℓ + Δ on graphs having a modulator M whose deletion leaves a graph of maximum degree Δ. We complement these result by showing that for parameter ℓ alone both problems become W[1]-hard already on 2-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. Finally, we show that using such families one can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem parameterized by k on both degenerate and nowhere dense classes of graphs.
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