On the strong metric dimension of antiprism graph, king graph, and graph

Let G be a connected graph with a set of vertices V(G) and a set of edges E(G). The interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u-v path. A vertex s ∊ V(G) is said to be strongly resolved for vertices u, v ∊ V(G) if v ∊ I[u, s] or u ∊ I[v, s]. A vertex set S ⊆ V(G) is a strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertices of S. The strong metric dimension of G, denoted by sdim(G), is defined as the smallest cardinality of a strong resolving set. In this paper, we determine the strong metric dimension of an antiprism An graph, a king Km,n graph, and a \({K}_{m}\odot {K}_{n}\) graph. We obtain the strong metric dimension of an antiprim graph An are n for n odd and n + 1 for n even. The strong metric dimension of King graph Km,n is m + n − 1. The strong metric dimension of \({K}_{m}\odot {K}_{n}\) graph are n for m = 1, n > 1 and mn − 1 for m > 2, n > 1.