On some results of perfect dominations of some graphs

Abstract

A dominating set D ⊆ V (G) of a simple graph G is the set of all u such that for every v ∈ V (G)\D, uv ∈ E(G). An independent set I ⊆ V (G) is a set of non-adjacent vertices in G. An independent dominating set Di ⊆ V (G) is a subset of V (G) that is both independent set and dominating set. A subset S of V (G) is called a perfect dominating set of S if for each v belongs to V (G)\S, there exists a unique element u ∈ S, such that v and u are adjacent. Define an independent perfect dominating set Dip of G to be a dominating set that is both independent dominating set and perfect dominating set. The minimum cardinality of an independent perfect dominating set of G is called an independent perfect domination number of G, denoted by γip(G). If a graph has a perfect dominating set, we say that the graph G is γip-graph. In this study, we determine some bounds and parameters of the graph as well as the existence existence of this invariant to some graphs and graphs formed by some binary operations.