Maximal point set domination in graphs

Abstract

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text] A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if for any [Formula: see text] in [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text] are adjacent. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text] Several domination parameters have been introduced. Among them, point set domination was conceived by Sampathkumar and Pushpa Latha [E. Sampathkumar and L. Pushpa Latha, Point-set domination number of a graph, Indian J. Pure Appl. Math. 24(4) (1993) 225–229]. A subset [Formula: see text] of [Formula: see text] is called a point set dominating set of [Formula: see text] if for every subset [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] is connected in [Formula: see text] The minimum cardinality of a point set dominating set of [Formula: see text] is called the point set domination number of [Formula: see text] and is denoted by [Formula: see text] Kulli [V. R. Kulli, The Maximal Domination Number of a Graph, Graph Theory Notes of New York XXXIII (Academy of Sciences, 1997), pp. 11–13] introduced the concept of maximal domination number in a graph. A subset [Formula: see text] of [Formula: see text] is called a maximal dominating set of [Formula: see text] if [Formula: see text] is a dominating set of [Formula: see text] and [Formula: see text] is not a dominating set of [Formula: see text] The minimum cardinality of a maximal dominating set of [Formula: see text] is denoted by [Formula: see text] In this paper, the concept of maximal domination is combined with point set domination and maximal point set domination is studied.