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 s d i m ( C m ⋄ P n ) = m ( n + 1 ) − m 2 − 1 for m ≥ 4 , n = 2 , m even; s d i m ( C m ⋄ P n ) = m ( n + 1 ) − m 2 − 2 for m ≥ 4 , n = 3 , m even; s d i m ( C m ⋄ P n ) = m ( n + 1 ) − 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.