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.

翻译：图$G$的支配集$S$被称为$r$-分组支配集，若$S$可划分为$S_1,S_2,\ldots,S_k$，使得每个单元$S_i$的大小为$r$且$G$中由$S_i$诱导的子图是连通的。$r$-分组支配集的概念推广了多种具有连通分量大小要求的支配集变体，包括普通支配集（$r=1$）、配对支配集（$r=2$）和连通支配集（$r$任意且$k=1$）。本文研究$r$-分组支配集问题的计算复杂性，即判定给定图是否存在一个至多包含$k$个单元的$r$-分组支配集。对于一般$r$，该问题在多种意义下难以求解，因其继承了连通支配集的难解性。因此我们关注$r$为常数或参数的情形，但发现对于任意固定$r>0$，该问题仍难以求解。基于此难解性，我们考虑关于常见图结构参数的参数化复杂度。首先发现当$r$和树宽为参数时问题是固定参数可解的，因为常数$r$下的$r$-分组支配条件可用一元二阶逻辑（mso2）表示。这虽是好消息，但运行时间并不实用。进而针对一般$r\ge 2$设计了一个$O^*(\min\{(2\tau(r+1))^{\tau},(2\tau)^{2\tau}\})$时间算法，其中$\tau$为孪生覆盖数——介于顶点覆盖数和团宽度之间的参数。对于配对支配集和三重组支配集（即$r\in\{2,3\}$），我们可加速算法，运行时间降至$O^*((r+1)^\tau)$。最后进一步探讨了FPT结果与图参数之间的关系，勾勒出$r$-分组支配集的参数化复杂度全景。