k-Hop Domination Defect in a Graph

Abstract

In this paper, we introduce a new graph parameter called the hop domination defect and  investigate it for some classes of graphs. The hop domination number $\gamma_h(G)$ of a graph $G$ is the minimum number of vertices required to hop dominate all the vertices of $G$. The minimality of $\gamma_h(G)$ implies that if $W \subseteq V(G)$ and $|W| < \gamma_h (G)$, then there is at least one vertex in $G$ that is not hop dominated by $W$. Given a positive integer $k < \gamma_h(G)$, where $\gamma_h(G) \geq 2$, the $k$-hop domination defect of $G$, denoted by $\zeta_k^h(G)$, is the minimum number of vertices of $G$ that is not hop dominated by any subset of vertices of $G$ with cardinality $\gamma_h(G) - k$. We give some bounds on the $k$-hop domination defect of a graph in terms of its order and maximum hop degree. Furthermore, we determine the $k$-hop domination defects of the join of some graphs.