On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets
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In a reconfiguration version of an optimization problem Q the input is an instance of Q and two feasible solutions S and T. The objective is to determine whether there exists a step-by-step transformation between S and T such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the Connected Dominating Set Reconfiguration problem (CDS-R). It was shown in previous work that the Dominating Set Reconfiguration problem (DS-R) parameterized by k, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique K_d,d as a subgraph, for some constant d ≥ 1. We show that the additional connectivity constraint makes the problem much harder, namely, that CDS-R is W[1]-hard parameterized by k+ℓ, the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on 5-degenerate graphs. On the positive side, we show that CDS-R parameterized by k is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs.
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