(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let \( G = (V(G),E(G)) \) be a simple graph. A perfect dominating set \( J V(G) \) is called a certified perfect dominating set of \( G \) if each vertex \( a J \) has either no neighbors or at least two neighbors in \( V(G) J \). The certified perfect domination number of \( G \) \( γ_cerp(G) \) represents the smallest size of a certified perfect dominating set in \( G \). A certified perfect dominating set of \( G \) that attains this minimum size, i.e., |J|=γ_cep(G) is referred to as a \( γ_cerp \)-set. In this paper, we first present some upper bounds for the certified perfect domination number of \( G \), investigate the relationship between certified domination and certified perfect domination parameters, and determine graphs with small and large values of these parameters. Secondly, we characterize the graphs with \( γ_cerp(G) = n \) and \( γ_cerp(G) = γ_cer(G) \). Lastly, we characterize the certified perfect dominating set under the lexicographic and Cartesian products of two graphs, determine its certified perfect domination number, and identify a non-\(γ_cerp\)-graph under these binary operations.