(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G = (V(G),E(G)) be a simple undirected graph. If γ_g(G) is the geodetic domination number of G and S V(G) such that |S| < γ_g(G), then definitely, there is at least one vertex of G that is not geodetically dominated by S, that is, not dominated by any vertex in S or not in any geodesic of any two vertices in S. If k is a positive integer with k γ_g(G) - 1 and S V(G) with |S| =γ_g(G) - k, then the number ζ_k^g(S) given by ζ_k^g(S) = |V(G) N_G^g[S]|, where N_G^g[S] = N_G[S] I_G[S], is called the k-geodetic domination defect of S in G. The k-geodetic domination defect of G is denoted and given by ζ_k^g(G) = {ζ_k^g(S): S V(G) and |S| = γ_g(G) - k}. In this paper, we study this newly defined parameter for some known classes of graphs. Moreover, we determine some sharp bounds of the parameter.