On the metric representation of the vertices of a graph
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912072417148928 |
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| author | Mora, Mercè Puertas, María Luz |
| author_facet | Mora, Mercè Puertas, María Luz |
| contents | The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{ω_1, \dots , ω_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,ω_1), \dots , d(u,ω_n))$. A vertex subset $W$ is a resolving set of $G$ if $r(u\vert W)\neq r(v\vert W)$, for every $u,v\in V(G)$ with $u\neq v$. Thus, a resolving set with $n$ elements provides a set of metric representation vectors $S\subset \mathbb{Z}^n$ with cardinal equal to the order of the graph. In this paper, we address the reverse point of view, that is, we characterize the finite subsets $S\subset \mathbb{Z}^n$ that are realizable as the set of metric representation vectors of a graph $G$ with respect to some resolving set $W$. We also explore the role that the strong product of paths plays in this context. Moreover, in the case $n=2$, we characterize the sets $S\subset \mathbb{Z}^2$ that are uniquely realizable as the set of metric representation vectors of a graph $G$ with respect to a resolving set $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the metric representation of the vertices of a graph Mora, Mercè Puertas, María Luz Combinatorics 05C12, 05C62 The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{ω_1, \dots , ω_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,ω_1), \dots , d(u,ω_n))$. A vertex subset $W$ is a resolving set of $G$ if $r(u\vert W)\neq r(v\vert W)$, for every $u,v\in V(G)$ with $u\neq v$. Thus, a resolving set with $n$ elements provides a set of metric representation vectors $S\subset \mathbb{Z}^n$ with cardinal equal to the order of the graph. In this paper, we address the reverse point of view, that is, we characterize the finite subsets $S\subset \mathbb{Z}^n$ that are realizable as the set of metric representation vectors of a graph $G$ with respect to some resolving set $W$. We also explore the role that the strong product of paths plays in this context. Moreover, in the case $n=2$, we characterize the sets $S\subset \mathbb{Z}^2$ that are uniquely realizable as the set of metric representation vectors of a graph $G$ with respect to a resolving set $W$. |
| title | On the metric representation of the vertices of a graph |
| topic | Combinatorics 05C12, 05C62 |
| url | https://arxiv.org/abs/2410.10411 |