3 research outputs found
Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond
A dominating set of graph is called an -grouped dominating set if
can be partitioned into such that the size of each
unit is and the subgraph of induced by is connected. The
concept of -grouped dominating sets generalizes several well-studied
variants of dominating sets with requirements for connected component sizes,
such as the ordinary dominating sets (), paired dominating sets (),
and connected dominating sets ( is arbitrary and ). In this paper, we
investigate the computational complexity of -Grouped Dominating Set, which
is the problem of deciding whether a given graph has an -grouped dominating
set with at most units. For general , 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 is a constant or a parameter,
but we see that the problem for every fixed 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 and treewidth, because the condition of -grouped domination for a
constant can be represented as monadic second-order logic (mso2). This is
good news, but the running time is not practical. We then design an
-time algorithm for general
, where 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., , we can speed up the algorithm, whose
running time becomes . We further argue the relationship
between FPT results and graph parameters, which draws the parameterized
complexity landscape of -Grouped Dominating Set.Comment: 23 pages, 6 figure