All 2-connected in-tournaments that are cycle complementary

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. A digraph DD is cycle complementary if there exist two vertex-disjoint directed cycles spanning the vertex set of DD. Let DD be a 2-connected in-tournament of order at least 8. In this paper we show that DD is not cycle complementary if and only if it is 2-regular and has odd order.