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 - Funding Information:
The publication of this paper is funded by Center for Higher Education Fund (Balai Pembiayaan Pendidikan Tinggi), Center of Education Services (Pusat Layanan Pendidikan), and Education Fund Management Institution (LPDP), Ministry of Education, Culture, Research, and Technology of the Republic of Indonesia.
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 -