On the 2-Linkage Problem for Semicomplete Digraphs

Abstract

We consider the problem of finding two disjoint directed paths with prescribed ends in a semicomplete digraph. The problem is NP - complete for general digraphs as proved in [4]. We obtain best possible sufficient conditions in terms of connectivity for a semicomplete digraph to be 2-linked (i.e., it contains two disjoint paths with prescribed ends for any four given endvertices). We also consider the algorithmically equivalent problem of finding a cycle through two given disjoint edges in a semicomplete digraph. For this problem it is shown that if D is a 5–connected semicomplete digraph, then D has a cycle through any two disjoint edges, and this result is best possible in terms of the connectivity. In contrast to this we prove that if T is a 3–connected tournament, then T has a cycle through any two disjoint edges. This is best possible, too. Finally we give best possible sufficient conditions in terms of local connectivities for a tournament to contain a cycle through af given pair of disjoint edges.