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

TL;DR

This paper studies the computational complexity of r-grouped dominating sets in graphs, exploring fixed-parameter tractability with respect to parameters like treewidth and twin cover, and providing algorithms for various cases.

Contribution

It introduces the concept of r-grouped dominating sets, analyzes their complexity, and develops fixed-parameter algorithms based on graph parameters such as twin cover.

Findings

Problem is hard for fixed r>0 due to inherited hardness from connected dominating set.

Fixed-parameter tractability established for r and treewidth via monadic second-order logic.

Developed an efficient algorithm with time complexity depending on twin cover number for general r.

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 $r$, 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 $O^*(\min\{(2\tau(r+1))^{\tau},(2\tau)^{2\tau}\})$-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.