Strong subgraph 2-arc-connectivity and arc-strong connectivity of Cartesian product of digraphs

Dong, Yiling ; Gutin, Gregory ; Sun, Yuefang 

Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 913-931

Résumé

Let D=(V,A) be a digraph of order n, S a subset of V of size k and 2≤ k≤ n. A strong subgraph H of D is called an S-strong subgraph if S⊆ V(H). A pair of S-strong subgraphs D_1 and D_2 are said to be arc-disjoint if A(D_1)∩ A(D_2)=∅. Let λ_S(D) be the maximum number of pairwise arc-disjoint S-strong subgraphs in D. The strong subgraph k-arc-connectivity is defined as λ_k(D)=min{λ_S(D)| S⊆ V(D), |S|=k}.The parameter λ_k(D) can be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we first obtain a formula for the arc-connectivity of Cartesian product λ(G□ H) of two digraphs G and H generalizing a formula for edge-connectivity of Cartesian product of two undirected graphs obtained by Xu and Yang (2006). Using this formula, we get a new formula for the arc-connectivity of Cartesion product of k≥ 2 copies of a strong digraph G: λ(G^k)=k·min{δ ^+ (G),δ ^- (G) }. Then we study the strong subgraph 2-arc-connectivity of Cartesian product λ_2(G□ H) and prove that min{λ (G) |H|, λ (H)|G|, δ ^+ (G)+ δ ^+ (H), δ ^- (G)+ δ ^- (H) }≥λ_2(G□ H)≥λ_2(G)+λ_2(H)-1. The upper bound for λ_2(G□ H) is sharp and is a simple corollary of the formula for λ(G□ H). The lower bound for λ_2(G□ H) is either sharp or almost sharp i.e., differs by 1 from the sharp bound. We improve the lower bound under an additional condition and prove its sharpness by showing that λ_2(G□ H)≥λ_2(G)+ λ_2(H), where G and H are two strong digraphs such that δ ^+ (H ) gt; λ _2(H). We also obtain exact values for λ_2(G□ H), where G and H are digraphs from some digraph families.

Détail

Keywords: connectivity, strong subgraph arc-connectivity, Cartesian product, tree connectivity