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A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interesting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, w e show that every strongly connected quasi-transitive digraph D on at least four vertices has two vertices u1 and u2 such that D -u, is strongly connected for i = 1,2. A result of tournaments on the existence of a pair of arc-disjoint in-and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore w e characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 I k I IV(D)l -1 through every vertex and yet they are not hamiltonian. Finally w e characterize pancyclic and vertex pancyclic quasi-transitive digraphs. 0 1995, John Wiley & Sons, Inc.

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Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satis es local edgeconnectivity prescriptions. An extension of Edmonds' theorem on disjoint arborescences is also deduced along with a new su cient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.