Total Safe Domination on Some Known Families of Graphs

Let G = (V (G),E(G)) be a connected simple graph. A total dominating set in G is a nonempty set S ⊆ V (G) such that every vertex v ∈ V (G) is adjacent to at least one vertex in S. A safe dominating set in G is a nonempty set S ⊆ V (G) which is a dominating set such that every component of G[S] is at least as large as any adjacent component of G[V (G) ∖ S]. This study introduces the concept of total safe domination in graphs, a combination of total domination and safe domination in graphs. Total safe domination ensures that no vertex in the dominating set is isolated and that every component of the subgraph induced by the set is robust, with respect to the adjacent components in the induced subgraph of the complement set. This guarantees both coverage and structural resilience. This paper presents the characterization of total safe dominating sets on some known graph families, namely: path, cycle, complete, sunlet, friendship, and helm graphs. It also determines the total safe domination number of the mentioned graphs.