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

zbMATH Open1463.05396arXiv1311.3804MaRDI QIDQ5206427

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Publication date: 18 December 2019

Abstract: Given a graph $G$ and a vertex $xinV(G)$, a vertex set $SsubseteqV(G)$ is an $x$-geodominating set of $G$ if each vertex $vinV(G)$ lies on an $x-y$ geodesic for some element $yinS$. 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 $partial(x)=vinV(G):forallwinN(v):d(x,w)leqd(u,v)$. 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.

Full work available at URL: https://arxiv.org/abs/1311.3804

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