Safe Sets in Some Graph Families

Abstract

For a connected simple graph G , a non-empty set \(S \subseteq V(G)\) of vertices is a safe set if, for every component \(A \text { of }\langle S\rangle_{G}\) and every component \(B \text { of }\langle V(G)-S\rangle_{G}\) adjacent to A , it holds that \(|A| \geq|B|\). The safe number denoted by s(G) of G is the minimum cardinality of a safe set G . In this paper, it examines the characterization of a safe set in complete bipartite graph. It also discusses the minimum cardinality of a safe sets of path graph and cycle graph via modulus. Moreover, this study generates the possible exact values of the safe number of the complete graph, complete bipartite graph, and star graph.