On the acyclic disconnection and the girth

The acyclic disconnection, (omega) over right arrow (D), of a digraph D is the maximum number of connected components of the underlying graph of D - A(D*), where D* is an acyclic subdigraph of D. We prove that (omega) over right arrow (D) >= g - 1 for every strongly connected digraph with girth g >= 4, and we show that (omega) over right arrow (D) = g - 1 if and only if D congruent to C-g for g >= 5. We also characterize the digraphs that satisfy (omega) over right arrow (D) = g - 1, for g = 4 in certain classes of digraphs. Finally, we define a family of bipartite tournaments based on projective planes and we prove that their acyclic disconnection is equal to 3. Then, these bipartite tournaments are counterexamples of the conjecture (omega) over right arrow (T) = 3 if and only if T congruent to (C) over right arrow (4) posed for bipartite tournaments by Figueroa et al. (2012). (C) 2015 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft

Bounds on the k-restricted arc connectivity of some bipartite tournaments

For k¿=¿2, a strongly connected digraph D is called -connected if it contains a set of arcs W such that contains at least k non-trivial strong components. The k-restricted arc connectivity of a digraph D was defined by Volkmann as . In this paper we bound for a family of bipartite tournaments T called projective bipartite tournaments. We also introduce a family of “good” bipartite oriented digraphs. For a good bipartite tournament T we prove that if the minimum degree of T is at least then where N is the order of the tournament. As a consequence, we derive better bounds for circulant bipartite tournaments.Peer ReviewedPostprint (author's final draft