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

THEORETICAL COMPUTER SCIENCE 2024年 996卷

作者： Hanaka, Tesshu Ono, Hirotaka Otachi, Yota Uda, Saeki Kyushu Univ Fukuoka Japan Nagoya Univ Nagoya Japan

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 is r and the subgraph of G induced by S 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{(2(r+1)),(2)(2)})-time algorithm for general r >= 2, where tau 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 is an element of {2,3}, we can speed up the algorithm, whose running time becomes O*((r+1)). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

关键词： Dominating set Paired dominating set Parameterized complexity graph structural parameters

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