A decomposition for digraphs with minimum outdegree 3 having no vertex disjoint cycles of different lengths

We say that a digraph $D=(V,A)$ admits a good decomposition $D=D_1\cup D_2\cup D_3$ if $D_1=(V_1,A_1), D_2=(V_2,A_2)$ and $D_3=(V_3,A_3)$ are such subdigraphs of $D$ that $V=V_1\cup V_2$ with $V_1\cap V_2=\emptyset$, $V_2\ne\emptyset$ but $V_1$ may be empty, $D_1$ is the subdigraph of $D$ induced by $V_1$ and is an acyclic digraph, $D_2$ is the subdigraph of $D$ induced by $V_2$ and is a strong digraph and $D_3$ is a subdigraph of $D$, every arc of which has its tail in $V_1$ and its head in $V_2$. In this paper, we show that a digraph $D=(V,A)$ with minimum outdegree 3 has no vertex disjoint directed cycles of different lengths if and only if $D$ admits a good decomposition $D=D_1\cup D_2\cup D_3$, where $D_1=(V_1,A_1), D_2=(V_2,A_2)$ and $D_3=(V_3,A_3)$ are such that $D_2$ has minimum outdegree 3 and no vertex disjoint directed cycles of different lengths and for every vertex $v\in V_1$, $d_{D_1\cup D_3}^+ (v)\ge 3$. Moreover, when such a good decomposition for $D$ exists, it is unique. By these results, the investigation of digraphs with minimum outdegree 3 having no vertex disjoint directed cycles of different lengths can be reduced to the investigation of strong such digraphs. Further, we classify strong digraphs with minimum outdegree 3 and girth 2 having no vertex disjoint directed cycles of different lengths.