The strong domination problem in block graphs and proper interval graphs

Abstract

In a graph [Formula: see text], the degree of a vertex [Formula: see text], denoted by [Formula: see text], is defined as the number of edges incident on [Formula: see text]. A set [Formula: see text] of vertices of [Formula: see text] is called a strong dominating set if for every [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text]. For a given graph [Formula: see text], Min-Strong-DS is the problem of finding a strong dominating set of minimum cardinality. The decision version of Min-Strong-DS is shown to be NP -complete for chordal graphs. In this paper, we present polynomial time algorithms for computing a strong dominating set in block graphs and proper interval graphs, two subclasses of chordal graphs. On the other hand, we show that for a graph [Formula: see text] with [Formula: see text]-vertices, Min-Strong-DS cannot be approximated within a factor of [Formula: see text] for every [Formula: see text], unless NP [Formula: see text] DTIME ([Formula: see text]). We also show that Min-Strong-DS is APX -complete for graphs with maximum degree [Formula: see text]. On the positive side, we show that Min-Strong-DS can be approximated within a factor of [Formula: see text] for graphs with maximum degree [Formula: see text].