TY - JOUR

T1 - A central local metric dimension on acyclic and grid graph

AU - Listiana, Yuni

AU - Susilowati, Liliek

AU - Slamin, Slamin

AU - Osaye, Fadekemi Janet

N1 - Publisher Copyright:
© 2023 the Author(s), licensee AIMS Press.

PY - 2023

Y1 - 2023

N2 - The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let G be a connected graph and V(G) be a vertex set of G. For an ordered set W = {x1, x2, …, xk } ⊆ V(G), the representation of a vertex x with respect to W is rG (x|W) = {(d(x, x1), d(x, x2), …, d(x, xk)}. The set W is said to be a local metric set of G if r(x|W) ≠ r(y|W) for every pair of adjacent vertices x and y in G. The eccentricity of a vertex x is the maximum distance between x and all other vertices in G. Among all vertices in G, the smallest eccentricity is called the radius of G and a vertex whose eccentricity equals the radius is called a central vertex of G. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set W is a central local metric set if W is a local metric set and contains all central vertices of G. The minimum cardinality of a central local metric set is called a central local metric dimension of G. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph.

AB - The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let G be a connected graph and V(G) be a vertex set of G. For an ordered set W = {x1, x2, …, xk } ⊆ V(G), the representation of a vertex x with respect to W is rG (x|W) = {(d(x, x1), d(x, x2), …, d(x, xk)}. The set W is said to be a local metric set of G if r(x|W) ≠ r(y|W) for every pair of adjacent vertices x and y in G. The eccentricity of a vertex x is the maximum distance between x and all other vertices in G. Among all vertices in G, the smallest eccentricity is called the radius of G and a vertex whose eccentricity equals the radius is called a central vertex of G. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set W is a central local metric set if W is a local metric set and contains all central vertices of G. The minimum cardinality of a central local metric set is called a central local metric dimension of G. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph.

KW - central local metric set

KW - central vertex

KW - diameter

KW - grid graphs

KW - radius

KW - trees

UR - http://www.scopus.com/inward/record.url?scp=85163884919&partnerID=8YFLogxK

U2 - 10.3934/math.20231085

DO - 10.3934/math.20231085

M3 - Article

AN - SCOPUS:85163884919

SN - 2473-6988

VL - 8

SP - 21298

EP - 21311

JO - AIMS Mathematics

JF - AIMS Mathematics

IS - 9

ER -