We say that a digraph _{1} ∪ _{2} ∪ _{3} if _{1} = (_{1}, _{1}), _{2} = (_{2}, _{2}) and _{3} = (_{3}, _{3}) are such subdigraphs of _{1} ∪ _{2} with _{1} ∩ _{2} = ∅, _{2} ≠ ∅ but _{1} may be empty, _{1} is the subdigraph of _{1} and is an acyclic digraph, _{2} is the subdigraph of _{2} and is a strong digraph and _{3} is a subdigraph of _{1} and its head in _{2}. In this paper, we show that a digraph _{1} ∪_{2} ∪_{3}, where _{1} = (_{1}, _{1}), _{2} = (_{2}, _{2}) and _{3} = (_{3}, _{3}) are such that _{2} has minimum outdegree 3 and no vertex disjoint directed cycles of different lengths and for every vertex _{1}, ^{+}_{D}_{1∪}_{D}_{3}(

#### Keywords

- digraph with minimum outdegree 3
- vertex disjoint cycles
- cycles of different lengths
- acyclic digraph
- strong digraph

#### MSC 2010

- 05C38
- 05C20
- 05C75

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