On Certified Perfect Domination in Graphs

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 \( G \) \( γcerp(G) \) 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|=γcep(G) is referred to as a \( γcerp \)-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 \( γcerp(G) = n \) and \( γcerp(G) = γcer(G) \). 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-\(γcerp\)-graph under these binary operations.