Abstract :
We consider two generalizations of tournaments, locally semicomplete digraphs introduced in Bang-Jensen (1990) and quasi-transitive digraphs introduced in Bang-Jensen and Huang (1995). We show that results by Thomassen (1984) on linkings in highly connected tournaments are also valid for these much larger classes of digraphs. We describe a polynomial algorithm for the 2-linkage problem in quasi-transitive digraphs. We do this by reducing the problem to the case of semicomplete digraphs for which the problem was solved in Bang-Jensen and Thomassen (1992). We obtain best possible sufficient conditions in terms of connectivity for a quasi-transitive digraph to be 2-linked as well as for a quasi-transitive digraph to have a cycle through two given arcs. Finally, we point out that some of our results are valid for classes of digraphs that are much more general than locally semicomplete digraphs and quasi-transitive digraphs. In particular, there is a polynomial algorithm for the 2-linkage problem for all digraphs that can be obtained from strong semicomplete digraphs by substituting arbitrary digraphs for vertices.
