Dominating sets reconfiguration under token sliding
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Let G be a graph and D_s and D_t be two dominating sets of G of size k. Does there exist a sequence 〈 D_0 = D_s, D_1, ..., D_ℓ-1, D_ℓ = D_t〉 of dominating sets of G such that D_i+1 can be obtained from D_i by replacing one vertex with one of its neighbors? In this paper, we investigate the complexity of this decision problem. We first prove that this problem is PSPACE-complete, even when restricted to split, bipartite or bounded tree-width graphs. On the other hand, we prove that it can be solved in polynomial time on dually chordal graphs (a superclass of both trees and interval graphs) or cographs.
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