Geodetically Undominated Vertices in a Graph

Let $G = (V(G),E(G))$ be a simple undirected graph. If $\gamma_g(G)$ is the geodetic domination number of $G$ and $S \subseteq V(G)$ such that $|S| < \gamma_{g}(G)$, 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 $k \le \gamma_g(G) - 1$ and $S \subseteq V(G)$ with $|S| =\gamma_{g}(G) - k$, then the number $\zeta_k^g(S)$ given by $\zeta_k^g(S) = |V(G) \setminus N_G^{g}[S]|$, where $N_G^{g}[S] = N_G[S] \cap I_G[S]$, is called the $k$-geodetic domination defect of $S$ in $G$. The $k$-geodetic domination defect of $G$ is denoted and given by $\zeta_k^{g}(G) = \min \{\zeta_k^g(S): S \subseteq V(G) \ \text{and} \ |S| = \gamma_{g}(G) - k\}$. In this paper, we study this newly defined parameter for some known classes of graphs. Moreover, we determine some sharp bounds of the parameter.