2,234 research outputs found
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Domination Problems in Nowhere-Dense Classes
We investigate the parameterized complexity of generalisations and variations
of the dominating set problem on classes of graphs that are nowhere dense. In
particular, we show that the distance- dominating-set problem, also known
as the -centres problem, is fixed-parameter tractable on any class that
is nowhere dense and closed under induced subgraphs. This generalises known
results about the dominating set problem on -minor free classes, classes
with locally excluded minors and classes of graphs of bounded expansion. A
key feature of our proof is that it is based simply on the fact that these
graph classes are uniformly quasi-wide, and does not rely on a structural
decomposition. Our result also establishes that the distance-
dominating-set problem is FPT on classes of bounded expansion, answering a
question of Ne{v s}et{v r}il and Ossona de Mendez
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