  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.
