MINIMALLY STRONG SUBGRAPH (k, l)-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 subset of V(H). Two S-strong subgraphs D-1 and D-2 are said to be arc-disjoint if A(D-1) boolean AND A(D-2) = empty set. Let lambda(S)(D) be the maximum number of arc-disjoint S-strong digraphs in D. The strong subgraph k-arc-connectivity is defined as lambda(k)(D) = min{lambda(S)(D) vertical bar S subset of V, vertical bar S vertical bar = k}. A digraph D = (V, A) is called minimally strong subgraph (k, l)-arc-connected if lambda(k)(D) >= l but for any arc e is an element of A, lambda(k) (D - e) <= l - 1. Let G(n, k, l) be the set of all minimally strong subgraph (k, l)-arc-connected digraphs with order n. We define G(n, k, l) = max{vertical bar A(D)vertical bar vertical bar D is an element of G(n, k, l)} and g(n, k, l) = min{vertical bar A(D)vertical bar vertical bar D is an element of G(n, k, l)}. In this paper, we study the minimally strong subgraph (k, l)-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, l) and G(n, k, l).