Certified Perfect Domination in Graphs

Let G = (V, E) be a simple connected graph. A set S subset of V (G) is called a certified perfect dominating set of G if every vertex nu is an element of V (G) \ S is dominated by exactly one element u is an element of S, such that u has either zero or at least two neighbors in V (G) \S. The minimum cardinality of a certified perfect dominating set of G is called the certified perfect domination number of G and denoted by gamma cerp(G). A certified perfect dominating set S of G with |S| = 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 and corona of graphs. Furthermore, the relationship between the perfect dominating set, and the certified perfect dominating set of a graph G are established.