On Distance and Strong Metric Dimension of the Modular Product
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911775119638528 |
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| author | Kang, Cong X. Kelenc, Aleksander Peterin, Iztok Yi, Eunjeong |
| author_facet | Kang, Cong X. Kelenc, Aleksander Peterin, Iztok Yi, Eunjeong |
| contents | The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of $G\diamond H$. This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07194 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Distance and Strong Metric Dimension of the Modular Product Kang, Cong X. Kelenc, Aleksander Peterin, Iztok Yi, Eunjeong Combinatorics 05C12, 05C76 The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of $G\diamond H$. This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs. |
| title | On Distance and Strong Metric Dimension of the Modular Product |
| topic | Combinatorics 05C12, 05C76 |
| url | https://arxiv.org/abs/2402.07194 |