The central local metric dimension of graphs with a single central vertex

A local metric set is a concept where the differences of two vertices on an edge of graph G can be identified. This concept is only relevant for a graph G, where all vertices is connected by an edge. Previous studies have examined the applications of local metric dimension, for instance applications in mathematical chemistry. A vertex in a graph is central if its greatest distance from any other vertex is the smallest as possible, a local metric set is even more interesting when it contains all the central vertices in the graph. This paper explores the development of a local metric set, namely the central local metric set. This concept also supports the affordable access to provide good services to the community in the placement of vital objects, such as health centers, education centers, and clean water stations, so that the community easy to access them. Let G with |V(G)|= n and W = {x1, x2,…,xk} ⊆ V(G) and k ≤ n. In G, x is represented with regard to W as follows r(x|W) = (d(x, x1), d(x, x2),…,d(x, xk)). If r(x|W) ≠ r(y|W), ∀x,y ∈V(G), and xy is the edge of G, then W is a local metric set of G. Let S(G) is a set of all central vertices of G. So, the central local metric set is W, if S(G) ⊆ W. Further, we called W with minimum cardinality as a central local basis set and the cardinality is called the central local metric dimension of G or lmds (G). This study explores the central local metric set of graph having only one central vertex, that is G ≅ K1 + H, for some cases of H. The results show that K1 + H, with |V(H)|= m, has only one central vertex if and only if no vertex in H has degree m-1. There are two possibilities of lmds (K1 + H), either the central set of K1 + H subset of its local basis set or no intersection between the central set and local basis set.