Some Properties of Central Local Metric Dimension on Corona Product Graph

Abstract

In this paper, we present an exploration of the central local metric dimension in corona product graphs. Thecentral local metric dimension is a new development concept of a local metric set containing all central vertices. In real life, there are many applications of local metric dimension and central vertices. If a vital object is represented as a vertex in a graph, then its placement can use the concept of a central vertex so that people can easily reach it. Suppose the vital objects are health services, education, and water stations. The government can use the concept of local metric dimension to optimize transportation infrastructure management and create good transportation governance for these vital objects. Suppose G is a connected graph with vertex set V (G) and order n. A central vertex in G is a vertex with the shortest distance to any other vertex in G. A central set, S(G), is a set whose elements are all the central vertices in G. Suppose W is a local metric set of G, W is a central local metric set of G if S(G) ⊆ W. If W is a local metric set with minimal cardinality, then |W| is the central local metric dimension of G. This paper presents some properties of the central local metric dimension of G ⊙ H. The results show that the elements of the central set of G ⊙ H are vertices in V (G ⊙ H) that coming from the central set ofG. Since in G ⊙ H, the i-th vertex of G adjacent to all vertices of i-th copies of H, then there is no intersection between the central set of G ⊙ H and the local metric set of G ⊙ H.