On the Metric Dimensions for Sets of Vertices
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866912570514866176 |
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| author | Hakanen, Anni Junnila, Ville Laihonen, Tero Puertas, María Luz |
| author_facet | Hakanen, Anni Junnila, Ville Laihonen, Tero Puertas, María Luz |
| contents | Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell\}$-resolving sets were recently introduced. In this paper, we present new results regarding the $\{\ell\}$-resolving sets of a graph. In addition to proving general results, we consider $\{2\}$-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of $\ell$-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for $\ell$-solid-resolving sets and show how $\ell$-solid- and $\{\ell\}$-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the $\ell$-solid- and $\{\ell\}$-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_02048 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Metric Dimensions for Sets of Vertices Hakanen, Anni Junnila, Ville Laihonen, Tero Puertas, María Luz Combinatorics Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell\}$-resolving sets were recently introduced. In this paper, we present new results regarding the $\{\ell\}$-resolving sets of a graph. In addition to proving general results, we consider $\{2\}$-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of $\ell$-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for $\ell$-solid-resolving sets and show how $\ell$-solid- and $\{\ell\}$-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the $\ell$-solid- and $\{\ell\}$-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach. |
| title | On the Metric Dimensions for Sets of Vertices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2003.02048 |