On the strong metric dimension of crossed prism graph and edge corona of cycle with path graph

Let G be a connected graph with vertex set V(G) and edge set E(G). The interval I[u, v] between u and v is defined as the collection of all vertices that belong to some shortest u − v path. A vertex s ∈ V(G) strongly resolves two vertices u and v if u belongs to a shortest v −s path, denoted by u ∈ I[v, s], or v belongs to a shortest u−s path, denoted by v ∈ I[u, s]. A set S ∈ V(G) is a strong resolving set if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension of G, denoted by sdim(G), is defined as the cardinality of the strong metric basis. In this research, we determine the strong metric dimension of crossed prism graph and edge corona graph C_m ◊ P_n. We obtain the strong metric dimension of crossed prism graph with n ≥ 4 is n for n even. The strong metric dimension of edge corona graph C_m ◊ P_n is $sdim(C_m\diamond P_n)=m(n+1)-\frac{m}{2}-1$ for m ≥ 4, n = 2, m even; $sdim(C_m\diamond P_n)=m(n+1)-\frac{m}{2}-2$for m ≥ 4, n = 3, m even; $sdim(C_m\diamond P_n)=m(n+1)-\frac{m}{2}-2$ for m ≥ 4, n ≥ 3, m even; sdim(C_m ◊P_n)=mn for m ≥ 3, n = 2, m odd; sdim(C_m ◊P_n) = mn−1 for m ≥ 3, n ≥ 3, m odd. (paper)