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.

Resolving set has several applications in the fields of science, engineering, and computer science. One application of the resolving set problem includes navigation robots, chemical structures, and supply chain management. Suppose the set $W=\left\{s_1,s_2,\dots ,s_k\right\}\subset V\left(G\right)$, the vertex representations of $x\in V\left(G\right)$ is $r_m\left(x\right|W)=\{d(x,s_1),d(x,s_2),\dots ,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m\left(v|W\right)\ne r_m\left(u\right|W)$ for $uv\in E\left(G\right)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $m{d}_l\left(G\right)$. In our paper, we determined the bounds of the local multiset dimension of the comb product of tree graphs.