Safe Sets in Some Graph Families
Klarice Shaira R. Tan
Mathematics Department, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, Philippines.
Isagani S. Cabahug, Jr. *
Mathematics Department, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, Philippines.
*Author to whom correspondence should be addressed.
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.
Keywords: Safe sets, safe number