Introducing Safe Domination in Graphs
Isagani S. Cabahug, Devine Fathy Mae S. Griño, Marsha Ella L. Maceren · International Journal of Mathematics Trends and Technology · 2023
For a nontrivial connected graph G with no isolated vertex, a nonempty subset S of the vertex set of G is a safe dominating set if and only if it is both secure and dominating. 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. In this paper, we extend the idea of safe and dominating sets by providing characterizations of the safe dominating sets of some graph families. In particular, this paper discusses the minimum cardinality of safe dominating sets of path, cycle, complete, and complete bipartite graphs.