Strong subgraph k-connectivity bounds

03/01/2018

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by   Yuefang Sun, et al.

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Let D=(V,A) be a digraph of order n, S a subset of V of size k and 2< k≤ n. Strong subgraphs D_1, ... , D_p containing S are said to be internally disjoint if V(D_i)∩ V(D_j)=S and A(D_i)∩ A(D_j)=∅ for all 1< i<j< p. Let κ_S(D) be the maximum number of internally disjoint strong digraphs containing S in D. The strong subgraph k-connectivity is defined as κ_k(D)={κ_S(D)| S⊆ V, |S|=k}. A digraph D=(V, A) is called minimally strong subgraph (k,ℓ)-connected if κ_k(D)≥ℓ but for any arc e∈ A, κ_k(D-e)≤ℓ-1. In this paper, we first give a sharp upper bound for the parameter κ_k(D) and then study the minimally strong subgraph (k,ℓ)-connected digraphs.