(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G = (V(G),E(G)) be a simple undirected graph. A set SV(G) is a geodetic hop dominating set in G if for every v V (G) S, there exist vertices x,y,z S such that d_G(x,v) = 2 and v lies in a y-z geodesic, that is, v I_G(y,z). The minimum cardinality of a geodetic hop dominating set of G, denoted by γ_hg(G), is called the geodetic hop domination number of G. The minimality of γ_hg(G) implies that if S V(G) such that |S| < γ_hg(G), then there is at least one vertex of G that is not geodetically hop-dominated by S. The k-geodetic hop domination defect of G, denoted by ζ_k^hg(G), is the minimum number of vertices of G that is not geodetically hop-dominated by any subset of vertices of G with cardinality γ_hg(G) - k. A set S V(G) of cardinality γ_hg(G) - k for which |V(G) N_G^hg[S]| = ζ^hg_k(G), where N_G^hg[S] = N_G^2[S] I_G[S], is called a ζ^hg_k -set of G. In this paper, we initiate the study of the concept of k-geodetic hop domination defect of a non-trivial graph G and investigate it for some known classes of graphs.