On the acyclic disconnection and the girth

The acyclic disconnection  , ω⃗(D), of a digraph DD is the maximum number of connected components of the underlying graph of D−A(D∗)D−A(D∗), where D∗D∗ is an acyclic subdigraph of DD. We prove that ω⃗(D)≥g−1 for every strongly connected digraph with girth g≥4g≥4, and we show that ω⃗(D)=g−1 if and only if D≅CgD≅Cg for g≥5g≥5. We also characterize the digraphs that satisfy ω⃗(D)=g−1, for g=4g=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 33. Then, these bipartite tournaments are counterexamples of the conjecture ω⃗(T)=3 if and only if T≅C⃗4 posed for bipartite tournaments by Figueroa et al. (2012).