Resolving sets tolerant to failures in three-dimensional grids
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929337269223424 |
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| author | Mora, Mercè Salorio, María José Souto Tarrío-Tobar, Ana Dorotea |
| author_facet | Mora, Mercè Salorio, María José Souto Tarrío-Tobar, Ana Dorotea |
| contents | An ordered set $S$ of vertices of a graph $G$ is a resolving set for $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any $k$ vertices from the set. This is equivalent to finding $(k+1)$-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differ in at least $k+1$ coordinates. This problem is also related with the study of the $(k+1)$-metric dimension of a graph, defined as the minimum cardinality of a $(k+1)$-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of $k\ge 1$ for which there exists a $(k+1)$-resolving set and construct such a resolving set of minimum cardinality in almost all cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_08768 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Resolving sets tolerant to failures in three-dimensional grids Mora, Mercè Salorio, María José Souto Tarrío-Tobar, Ana Dorotea Combinatorics 05C12, 05C76 An ordered set $S$ of vertices of a graph $G$ is a resolving set for $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any $k$ vertices from the set. This is equivalent to finding $(k+1)$-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differ in at least $k+1$ coordinates. This problem is also related with the study of the $(k+1)$-metric dimension of a graph, defined as the minimum cardinality of a $(k+1)$-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of $k\ge 1$ for which there exists a $(k+1)$-resolving set and construct such a resolving set of minimum cardinality in almost all cases. |
| title | Resolving sets tolerant to failures in three-dimensional grids |
| topic | Combinatorics 05C12, 05C76 |
| url | https://arxiv.org/abs/2112.08768 |