A central local metric dimension on acyclic and grid graph

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, \ldots, x_k\} \subseteq 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), \ldots, d(x, x_k) \} $. The set $ W $ is said to be a local metric set of $ G $ if $ r(x|W)\neq 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.