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On the boundary as an $x$-geodominating set in graphs

Given a graph $G$ and a vertex $x\in V(G)$, a vertex set $S \subseteq V(G)$ is an $x$-geodominating set of $G$ if each vertex $v\in V(G)$ lies on an $x-y$ geodesic for some element $y\in S$. The minimum cardinality of an $x$-geodominating set of $G$ is defined as the $x$-geodomination number of $G$, $g_x(G)$, and an $x$-geodominating set of cardinality $g_x(G)$ is called a $g_x$-set and it is known that it is unique for each vertex $x$. We prove that, in any graph $G$, the $g_x$-set associated to a vertex $x$ is the set of boundary vertices of $x$, that is $\partial(x)= \{v \in V(G) : \forall w \in N(v): d(x,w) \leq d(u, v)\}$. This characterization of $g_x$-sets allows to deduce, on a easy way, different properties of these sets and also to compute both $g_x$-sets and $x$-geodomination number $g_x(G)$, in graphs obtained using different graphs products: cartesian, strong and lexicographic.