Central Metric Dimension of Rooted Product Graph

The central metric dimension is a type of metric dimension on graph. Some special graphs for which the central metric dimension have been found include path graph, cycle graph, complete graph, and complete bipartite graph. The aim of this study is to determine the central metric dimension of rooted product graph. Let G be a connected graph of order n and H is a sequence of n rooted graphs H₁, H₂, H₃, …, H_n. The rooted product graph G and H denoted by G ◦ H. In this paper, we determine the central metric dimension of rooted product graph G ◦ H, which denoted by dimcen(G ◦ H). The results obtained for G ◦ H, where H is a sequence of rooted graphs that all have the same radius and the rooted vertex is the central vertex. For H is a sequence of rooted cycle graph, the cycle with the largest radius has an impact on the central set, while the central metric dimension is affected by the central set of G ◦ H. For H is a sequence of rooted complete graph, the central set is affected by the central set of a graph G, while the central metric dimension is affected by the central set of the graph G.