On Distance and Strong Metric Dimension of the Modular Product: On Distance and Strong Metric

The modular product G ⋄ H of graphs G and H is a graph on vertex set V ( G ) × V ( H ) . Two vertices ( g , h ) and ( g ′ , h ′ ) of G ⋄ H are adjacent if g = g ′ and h h ′ ∈ E ( H ) , or g g ′ ∈ E ( G ) and h = h ′ , or g g ′ ∈ E ( G ) and h h ′ ∈ E ( H ) , or (for g ≠ g ′ and h ≠ h ′ ) g g ′ ∉ E ( G ) and h h ′ ∉ E ( H ) . We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of G ⋄ H . This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs.