Tournament completions of bipartite tournaments and their augmented directed cycles

A tournament T is a tournament completion of a bipartite tournament D if D is a spanning subdigraph of T, i.e., $V(D)=V(T)$ and A(D)⊆ A(T). If C is a k-dicycle (i.e., directed cycle of length k) in a tournament completion T of D and C is not a dicycle in D, i.e., A(C)⊆ A(T) and A(C)⊆ A(D), then we call C an augmented k-dicycle of T. In this paper, we investigate the families of bipartite tournaments for which there exists a tournament completion with exactly one augmented $3$-dicycle and with no augmented $4$-dicycles. Our investigation may be viewed as a variant of the orientation completion problem initiated by Bang-Jensen et al..