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Journal of Discrete Mathematical Sciences and Cryptography

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

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Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

The central local metric dimension of graphs with a single central vertex

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pp. 2297–2308Vol. 28Issue 6September 2025DOI: 10.47974/JDMSC-2228 Crossmark XML
Received:
04 Jun 2024
Published Online:
12 Jul 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2228
Pages:
2297–2308

Abstract

A local metric set is a concept where the differences of two vertices on an edge of graph G can be identified. This concept is only relevant for a graph G, where all vertices is connected by an edge. Previous studies have examined the applications of local metric dimension, for instance applications in mathematical chemistry. A vertex in a graph is central if its greatest distance from any other vertex is the smallest as possible, a local metric set is even more interesting when it contains all the central vertices in the graph. This paper explores the development of a local metric set, namely the central local metric set. This concept also supports the affordable access to provide good services to the community in the placement of vital objects, such as health centers, education centers, and clean water stations, so that the community easy to access them. Let G with |V(G)|= n and W = {x1, x2,…,xk} ⊆ V(G) and k ≤ n. In G, x is represented with regard to W as follows r(x|W) = (d(x, x1), d(x, x2),…,d(x, xk)). If r(x|W) ≠ r(y|W), ∀x,y ∈V(G), and xy is the edge of G, then W is a local metric set of G. Let S(G) is a set of all central vertices of G. So, the central local metric set is W, if S(G) ⊆ W. Further, we called W with minimum cardinality as a central local basis set and the cardinality is called the central local metric dimension of G or lmds (G). This study explores the central local metric set of graph having only one central vertex, that is G ≅ K1 + H, for some cases of H. The results show that K1 + H, with |V(H)|= m, has only one central vertex if and only if no vertex in H has degree m-1. There are two possibilities of lmds (K1 + H), either the central set of K1 + H subset of its local basis set or no intersection between the central set and local basis set.

Keywords

Central local metric dimensionLocal metric setCentral vertexA single central vertex graphAffordable access

Subject Classifications

05C12

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