k-Hop Domination Defect in a Graph (European Journal of Pure and Applied Mathematics, 2025)
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 $γ_h(G)$ of a graph $G$ is the minimum number of vertices required to hop dominate all the vertices of $G$. The minimality of $γ_h(G)$ implies that if $W \subseteq V(G)$ and $|W| < γ_h (G)$, then there is at least one vertex in $G$ that is not hop dominated by $W$. Given a positive integer $k < γ_h(G)$, where $γ_h(G) ≥ 2$, the $k$-hop domination defect of $G$, denoted by $ζ_k^h(G)$, is the minimum number of vertices of $G$ that is not hop dominated by any subset of vertices of $G$ with cardinality $γ_h(G) - k$. 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.
Keywords: hop domination, k-hop domination defect, hop degree of a graph, join