Mutual-visibility in strong products of graphs via total mutual-visibility
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910414127759360 |
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| author | Cicerone, Serafino Di Stefano, Gabriele Klavžar, Sandi Yero, Ismael G. |
| author_facet | Cicerone, Serafino Di Stefano, Gabriele Klavžar, Sandi Yero, Ismael G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07835 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mutual-visibility in strong products of graphs via total mutual-visibility Cicerone, Serafino Di Stefano, Gabriele Klavžar, Sandi Yero, Ismael G. Combinatorics 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. |
| title | Mutual-visibility in strong products of graphs via total mutual-visibility |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2210.07835 |