(untitled) (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 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.