Maximal point set domination in graphs.

Let G = (V , E) be a graph with vertex set V and edge set E. A subset D of V is called a dominating set of G if for any v in V − D , there exists a vertex u ∈ D such that u and v are adjacent. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ (G). 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 D of V (G) is called a point set dominating set of G if for every subset T of V − D , there exists a vertex u ∈ D such that T ∪ { u } is connected in G. The minimum cardinality of a point set dominating set of G is called the point set domination number of G and is denoted by γ p (G). 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 D of V (G) is called a maximal dominating set of G if D is a dominating set of G and V − D is not a dominating set of G. The minimum cardinality of a maximal dominating set of G is denoted by γ m (G). In this paper, the concept of maximal domination is combined with point set domination and maximal point set domination is studied. [ABSTRACT FROM AUTHOR]