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Tournaments and Semicomplete Digraphs

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Classes of Directed Graphs

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Abstract

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.

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Notes

  1. 1.

    Note that here we allowed one of the two tournaments to be empty, in which case the corresponding path is also empty

  2. 2.

    See also [9, Section 9.1].

  3. 3.

    private communication, August 1999.

  4. 4.

    For simplicity and because they are polynomially equivalent, we do not distinguish between the decision and the optimization versions of these problems.

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  1. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK5230, Odense M, Denmark

    Jørgen Bang-Jensen

  2. Projet COATI, INRIA, Sophia-Antipolis, 2004 Route de Lucioles BP93, 06902, Sophia-Antipolis Cedex, France

    Frédéric Havet

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  2. Department of Computer Science, Royal Holloway, University of London, Egham, United Kingdom

    Gregory Gutin

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Bang-Jensen, J., Havet, F. (2018). Tournaments and Semicomplete Digraphs. In: Bang-Jensen, J., Gutin, G. (eds) Classes of Directed Graphs. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-71840-8_2

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