Abstract
The existence of a polynomial time algorithm for finding a cycle covering a given set of vertices in a semicomplete multipartite digraph (if it exists) was conjectured by J. Bang-Jensen, G. Gutin, and A. Yeo. The analog problem for semicomplete bipartite digraphs was conjectured by J. Bang-Jensen and Y. Manoussakis. We prove the conjecture of Bang-Jensen et al. in the affirmative, which also implies the second conjecture. We, furthermore, give a polynomial time algorithm to find a cycle covering a given set of vertices in a semicomplete bipartite digraph, which is the longest of all such cycles, and we conjecture that such an algorithm also exists for semicomplete multipartite digraphs.
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