The combination concept of locating and dominating set of rooted product graphs

The dominant metric dimension is an important aspect of graph theory, particularly concerning the ideas of locating sets and dominating sets. This research focuses on finding the dominant metric dimension of rooted product graphs formed from a connected graph G and sequence H. Let G be a connected graph of order n and H be a sequence of n rooted graph H1, H2, …, Hn. Rooted product graph G ∘ H can be constructed from the graphs G, H1, H2, …, Hn by identifying the rooted vertex of each Hi with the i −th vertex of graph G. The outcome of this study is the dominant metric dimension of any connected graph G alongside sequence H. For any sequence of rooted star graphs, the dominant metric dimension equates to the total of the dominant metric dimension of the Hi where i =1,2, …, n. For any sequence of rooted complete graphs, the overall dominant metric dimension is equal to the total of the dominant metric dimensions of the Hi minus one, i = 1,2, …, n.