Characterizing 2-Distance Certified Hop Dominating Sets Using Co-Certified Pointwise Non-domination Concept

Abstract

Let $G$ be a graph. Then $C \subseteq V(G)$ is called a 2-distance certified hop dominating if $\forall$ $x \in V(G) \backslash C$, there exists $y \in C$ such that $d_G(x,y)=2$ and $\forall$ $a \in C$, $|N^2_G (a)\backslash C|=0$ or $|N^2_G (a)\backslash C|\geq 2$. The 2-distance certified hop domination number of $G$, denoted by $\gamma_{2ch}(G)$, is the minimum cardinality among all 2-distance certified hop dominating sets of $G$. In this study, the researchers give some properties of this concept on some graphs, and present some its connections with others parameters. Also, the researchers have introduced co-certified pointwise non-domination to characterize the 2-distance certified hop dominating sets in the join of two graphs. They obtain some simplified formulas of the said parameter on the join of some graphs using this newly defined concept and some characterizations formulated.