Tournaments and Semicomplete Digraphs

The class of tournaments is by far the most well-studied class of digraphs with many deep and important results. Since Moon’s pioneering book in 1968 [146], the study of tournaments and their properties has flourished and research on tournaments is still a very active area. Often this research deals with the superclass of semicomplete digraphs which are digraphs with no pair of non-adjacent vertices (that is, contrary to tournaments, we allow directed cycles of length 2). In this chapter we cover a very broad range of results on tournaments and semicomplete digraphs from classical to very recent ones. In order to stimulate further research, we not only list a number of open problems, but also give a number of proofs which illustrate the diversity of proof techniques that have been applied. These range from elementary to quite advanced.