On lower bounds for the metric dimension of graphs
| Autor: | Mohsen Jannesari |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Journal of Mahani Mathematical Research, Vol 12, Iss 1, Pp 35-41 (2023) |
| Druh dokumentu: | article |
| ISSN: | 2251-7952 2645-4505 |
| DOI: | 10.22103/jmmrc.2022.19121.1211 |
| Popis: | For an ordered set $W=\{w_1, w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W)=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension, and a resolving set of minimum cardinality is a basis of $G$. Lower bounds for metric dimension are important. In this paper, we investigate lower bounds for metric dimension. Motivated by a lower bound for the metric dimension $k$ of a graph of order $n$ with diameter $d$ in [S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs, Discrete Applied Mathematics $70(3) (1996) 217-229$], which states that $k \geq n-d^k$, we characterize all graphs with this lower bound and obtain a new lower bound. This new bound is better than the previous one, for graphs with diameter more than $3$. |
| Databáze: | Directory of Open Access Journals |
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