Minimally strong subgraph (k,\ell)-arc-connected digraphs

Let D = (V,A) be a digraph of order n, S a subset of V of size k and 2 ≤ k ≤ n. A subdigraph H of D is called an S-strong subgraph if H is strong and S ⊆ V (H). Two S-strong subgraphs D1 and D2 are said to be arc-disjoint if A(D1) ∩ A(D2) = ∅. Let λS(D) be the maximum number of arc-disjoint S-strong digraphs in D. The strong subgraph karc-connectivity is defined as λk(D) = min{λS(D) | S ⊆ V, |S| = k}. A digraph D = (V,A) is called minimally strong subgraph (k, `)-arc-connected if λk(D) ≥ ` but for any arc e ∈ A, λk(D − e) ≤ ` − 1. Let G(n, k, `) be the set of all minimally strong subgraph (k, `)-arc-connected digraphs with order n. We define G(n, k, `) = max{|A(D)| | D ∈ G(n, k, `)} and g(n, k, `) = min{|A(D)| | D ∈ G(n, k, `)}. In this paper, we study the minimally strong subgraph (k, `)-arc-connected digraphs. We give a characterization of the minimally strong subgraph (3, n − 2)-arc-connected digraphs, and then give exact values and bounds for the functions g(n, k, `) and G(n, k, `).