Complementary cycles containing prescribed vertices in tournaments

We prove that if T is a tournament on n≥7 vertices and x, y are distinct vertices of T with the property that T remains 2-connected if we delete the arc between x and y, then there exist disjoint 3-cycles C_x, C_y such that x ∈ V(C_x) and y ∈ V(C_y). This is best possible in terms of the connectivity assumption. Using this result, we prove that under the same connectivity assumption and if n≥8, then T also contains complementary cycles C^l_x,C^l_y (i.e. V(C^l_x) ∪ V(C^l_y)= V(T) and V(C^l_x) ∩ V(C^l_y) = ∅) such that x ∈ V(C^l_x) and y ∈ V(C^l_y) for every choice of distinct vertices x, y ∈ V(T). Again this is best possible in terms of the connectivity assumption. It is a trivial consequence of our result that one can decide in polynomial time whether a given tournament T with special vertices x, y contains disjoint cycles C_x, C_y such that x ∈ V(C_x) and y C ∈ V(C_y). This problem is NP-complete for general digraphs and furthermore there is no degree of strong connectivity which suffices to guarantee such cycles in a general digraph.