Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond

Abstract

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into $S_1,S_2,\ldots ,S_k$ 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 ($\textsf{MSO}_{2}$). This fixed-parameter tractability is good news, but the running time is not practical. We then design an $O^*(\min \{(2\tau (r+1))^{\tau },(2\tau )^{2\tau }\})$-time algorithm for general $r\ge 2$, 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., $r \in \{2,3\}$, we can speed up the algorithm, whose running time becomes $O^*((r+1)^\tau )$. We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r -Grouped Dominating Set.

Partially supported by JSPS KAKENHI Grant Numbers JP17H01698, JP17K19960, JP18H04091, JP20H05793, JP20H05967, JP21K11752, JP21H05852, JP21K17707, JP21K19765, and JP22H00513.