Grouped domination parameterized by vertex cover, twin cover, and beyond

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S₁,S₂,…,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, r-GROUPED DOMINATING SET 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 r-GROUPED DOMINATING SET for every fixed r>0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-GROUPED DOMINATING SET is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO₂). This fixed-parameter tractability 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.