## Abstract

Let G be a connected graph. A subset S of V(G) is called a local metric generator for G if for every two adjacent vertices u and v of G there exists a vertex w∈ S such that d_{G}(u, w) ≠ d_{G}(v, w) where d_{G}(x, y) is the distance between vertices x and y in G. The local metric dimension of G, denoted by dim _{ℓ}(G) , is the minimum cardinality among all local metric generators of G. The clique number ω(G) of G is the cardinality of a maximum set of vertices that induce a complete graph in G. The authors in [Local metric dimension for graphs with small clique numbers. Discrete Math. 345 (2022), no. 4, Paper No. 112763] conjectured that if G is a connected graph of order n with ω(G) = k where 2 ≤ k≤ n, then dimℓ(G)≤(k-1k)n. In this paper, we prove this conjecture. Furthermore, we prove that equality in this bound is satisfied if and only if G is a complete graph K_{n}.

Original language | English |
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Article number | 5 |

Journal | Graphs and Combinatorics |

Volume | 39 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2023 |

## Keywords

- Clique
- Local metric dimension
- Local metric generator

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics