An optimal algorithm to find minimum k-hop dominating set of interval graphs

Abstract

For a fixed positive integer [Formula: see text], a [Formula: see text]-hop dominating set [Formula: see text] of a graph [Formula: see text] is a subset of [Formula: see text] such that every vertex [Formula: see text] is within [Formula: see text]-steps from at least one vertex [Formula: see text], i.e., [Formula: see text]. A [Formula: see text]-hop dominating set [Formula: see text] is said to be minimal if there does not exist any [Formula: see text] such that [Formula: see text] is a [Formula: see text]-hop dominating set of G. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop dominating set, if it is minimal as well as it is [Formula: see text]-hop dominating set. In this paper, we present an optimal algorithm to find a minimum [Formula: see text]-hop dominating set of interval graphs with [Formula: see text] vertices which runs in [Formula: see text] time.