$k$-Geodetic Hop Domination Defect in a Graph

Let $G = (V(G),E(G))$ be a simple undirected graph. A set $S\subseteq V(G)$ is a geodetic hop dominating set in $G$ if for every $v \in V (G) \setminus S$, there exist vertices $x,y,z \in S$ such that $d_G(x,v) = 2$ and $v$ lies in a $y$-$z$ geodesic, that is, $v \in I_G(y,z)$. The minimum cardinality of a geodetic hop dominating set of $G$, denoted by $\gamma_{hg}(G)$, is called the geodetic hop domination number of $G$. The minimality of $\gamma_{hg}(G)$ implies that if $S \subseteq V(G)$ such that $|S| < \gamma_{hg}(G)$, 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 $\zeta_k^{hg}(G)$, is the minimum number of vertices of $G$ that is not geodetically hop-dominated by any subset of vertices of $G$ with cardinality $\gamma_{hg}(G) - k$. A set $S \subseteq V(G)$ of cardinality $\gamma_{hg}(G) - k $ for which $|V(G) \setminus N_G^{hg}[S]| = \zeta^{hg}_{k}(G)$, where $N_G^{hg}[S] = N_G^{2}[S] \cap I_G[S]$, is called a $\zeta^{hg}_{k} $-\textit{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.