2 research outputs found
Lower Bounds on the Distance Domination Number of a Graph
For an integer , a (distance) -dominating set of a connected
graph is a set of vertices of such that every vertex of is at distance at most~ from some vertex of . The
-domination number, , of is the minimum cardinality of a
-dominating set of . In this paper, we establish lower bounds on the
-domination number of a graph in terms of its diameter, radius and girth. We
prove that for connected graphs and , , where denotes the direct product of
and
A Linear-Time Algorithm for Minimum -Hop Dominating Set of a Cactus Graph
Given a graph and an integer , a -hop dominating set
of is a subset of , such that, for every vertex , there
exists a node whose hop-distance from is at most . A -hop
dominating set of minimum cardinality is called a minimum -hop dominating
set. In this paper, we present linear-time algorithms that find a minimum
-hop dominating set in unicyclic and cactus graphs. To achieve this, we show
that the -dominating set problem on unicycle graph reduces to the piercing
circular arcs problem, and show a linear-time algorithm for piercing sorted
circular arcs, which improves the best known -time algorithm