Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond
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A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S_1,S_2,…,S_k such that the size of each unit S_i is r and the subgraph of G induced by S_i is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1), paired dominating sets (r=2), and connected dominating sets (r is arbitrary and k=1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, the problem is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that the problem for every fixed r>0 is still hard to solve. From the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that it is fixed-parameter tractable for r and treewidth, because the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (mso2). This is good news, but the running time is not practical. We then design an O^*(min{(2τ(r+1))^τ,(2τ)^2τ})-time algorithm for general r≥ 2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r ∈{2,3}, we can speed up the algorithm, whose running time becomes O^*((r+1)^τ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.
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