The reliability of lexicographic product digraphs

Abstract

For two digraphs D=(V1,A1) and H=(V2,A2), the lexicographic product digraph D[H] is the digraph with vertex set V1 × V2. There is an arc from vertex (x1, y1) to vertex (x2, y2) in D[H] if and only if either x1x2 ∈ A1 or x1=x2 and y1y2 ∈ A2. The minimum degree and the arc-connectivity of D are denoted by δ(D) and λ(D), respectively. We prove that for any two digraphs D and H, λ(D[H])≥min{n(t+λ(D)−δ(D)−1)+δ(D[H]),n2λ(D)} holds for any t≤min{δ(D)−λ(D)+1,λ(D)+1,n−1}, where n=|V(H)|. As a consequence, λ(D[H])≥n(λ(D)−δ(D))+δ(D[H]). We also provide some sufficient conditions for D[H] to have maximum reliability with respected the connectedness and super connectedness.