Secure Dominant Metric Dimension of New Classes of Graphs

Assume that G = (V, E) is a finite, connected, basic graph. If, for every y ∈ V(G), the code of y with respect to T, represented by CT (y), which is defined as CT (y) = (d(u1, y), d(u2, y),.., d(uk, y), is different for distinct y, then the subset T = {u1, u2,..., uk}of V(G) is said to be a resolving set. Dim (G) represents the dimension of G, which is the least cardinality of a resolving set. The resolving set S is secure if, for every t ∈ V – S, there exists r ∈ S such that (S – {r}) ⋃ {t} is a resolving set. The secure metric dimension of 𝐺 is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. Additionally, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure dominant metric dimension of special graphs such as triangular snake graph, middle graph, bistar graph, square of bistar graph and coconut tree. Finally, we derive the explicit formulas for the secure metric dimension of line graph, fan graph, diamond graph, subdivision of globe graph, Y- tree, F- tree, n- centipede tree and coconut tree.