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 $W=\{x_1,x_2,\dots ,x_k\}\subseteq V(G)$, the representation of a vertex $x$ with respect to $W$ is $r_G(x|W)=\{(d(x,x_1),d(x,x_2),\dots ,d(x,x_k)\}$. The set $W$ is said to be a local metric set of $G$ if $r(x|W)\ne r(y|W)$ 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.