Mutual-visibility in strong products of graphs via total mutual-visibility
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 $\mu(G)$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.07835
- arXiv:
- arXiv:2210.07835
- Bibcode:
- 2022arXiv221007835C
- Keywords:
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- Mathematics - Combinatorics