Conditions of Safe Dominating Set in Some Graph Families

Let X be an arbitrary Banach space. For a nontrivial connected graph G and nonempty subset S \(\subseteq\) V (G), S is a safe dominating set of G if and only if S is a dominating set of G and every component X of G[S] and every component Y of G[V (G) \ S] adjacent to X, |X| \(\ge\) |Y|. Moreover, S is called a minimum safe dominating set if S is a safe dominating set of the smallest size in a given graph. The cardinality of the minimum safe dominating set of G is the safe domination number of G, denoted by \(\gamma s\)(G). In this paper, we characterized the safe dominating set and determine its corresponding safe domination number in some special classes of graphs.