@article{Listiana2023, 
author = {Yuni Listiana and Liliek Susilowati and Slamin Slamin and Fadekemi Janet Osaye},
title = {A central local metric dimension on acyclic and grid graph},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {21298-21311},
keywords = {central local metric set, central vertex, diameter, radius, trees, grid graphs},
url = {https://www.sciopen.com/article/10.3934/math.20231085},
doi = {10.3934/math.20231085},
abstract = {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  =  {      x    1    ,      x    2    ,  …  ,      x    k    }  ⊆  V  (  G  ), the representation of a vertex    x with respect to    W is        r    G    (  x      |    W  )  =  {  (  d  (  x  ,      x    1    )  ,  d  (  x  ,      x    2    )  ,  …  ,  d  (  x  ,      x    k    )  }. 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.}
}