Certified Perfect Domination in Graphs

Let $ G = (V(G),E(G)) $ be a simple connected graph. A set $ S \subseteq V(G) $ is called a certified perfect dominating set of $ G $ if every vertex $ v \in V(G)\setminus S $ is dominated by exactly one element $ u \in S $, such that $ u $ has either zero or at least two neighbors in $ V(G)\setminus S $. The minimum cardinality of a certified perfect dominating set of $ G $ is called the \textit{certified perfect domination number} of $ G $ and denoted by $ \gamma_{cerp}(G) $. A certified perfect dominating set $ S $ of $ G $ with $ \lvert S \rvert = \gamma_{cerp}(G) $ is called a $ \gamma_{cerp} $-set. In this paper, the author focuses on several key aspects: a characterization of the certified perfect dominating set, determining the exact values of the certified perfect domination number for specific graphs, and investigating the certified perfect domination number of graphs resulting from the join of two graphs. Furthermore, some relationships between the certified dominating set, the perfect dominating set, and the certified perfect dominating set of a graph $ G $ are established.