Super connectivity of direct product of graphs

Abstract

For a graph G , κ ( G ) denotes its connectivity. A graph G is super connected , or simply super- κ , if every minimum separating set is the neighborhood of a vertex of G , that is, every minimum separating set isolates a vertex. The direct product G 1 × G 2 of two graphs G 1 and G 2 is a graph with vertex set V ( G 1 × G 2 ) = V ( G 1 ) × V ( G 2 ) and edge set E ( G 1 × G 2 ) = {( u 1 , v 1 )( u 2 , v 2 ) ∣ u 1 u 2 ∈ E ( G 1 ), v 1 v 2 ∈ E ( G 2 )} . Let Γ = G × K n , where G is a non-trivial graph and K n ( n ≥ 3) is a complete graph on n vertices. In this paper, we show that Γ is not super- κ if and only if either κ (Γ ) = n κ ( G ) , or Γ ≅ K ℓ, ℓ × K 3 (ℓ > 0) .