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arXiv · 2210.07835

Mutual-visibility in strong products of graphs via total mutual-visibility

Serafino Cicerone·Gabriele Di Stefano·Sandi Klavžar·Ismael G. Yero
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Abstract

Let $G$ be a graph and $X\subseteq V(G)$. Then $X$ is a mutual-visibility set if each pair of vertices from $X$ is connected by a geodesic with no internal vertex in $X$. The mutual-visibility number $μ(G)$ of $G$ is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.

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Serafino Cicerone, Gabriele Di Stefano, Sandi Klavžar, Ismael G. Yero. 2024-04-18. Mutual-visibility in strong products of graphs via total mutual-visibility. https://arxiv.org/abs/2210.07835

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