Weak and strong domination in thorn graphs

Abstract

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.