Non simple tournaments : Theoretical properties and a polynomial algorithm

  • A. Astie-Vidal
  • A. Matteo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 356)


The first part of this paper is a theoretical study of the properties of the convex subsets of a tournament.

The main new results are about regular tournaments :
  • Theorem 2 computes the maximum cardinality of a convex subset of a regular tournament.

  • Theorem 3 gives the structure of a regular tournament of order 3k with a convex subset of cardinality k.

  • Theorem 4 computes the maximum number of convex subsets of cardinality m that may be contained in a regular tournament of order n.

In the second part, we use these results to write an 0(n3) — algorithm which allows us to determine all the convex subsets of a regular tournament (this algorithm gives then, in polynomial-time, the answer to the following question : is a regular tournament simple or not ?).

The results of [2] about the decomposition of directed graphs, applied to regular tournaments allow to determine their convex subsets in time 0(n4).

Very recent results of [1] about digraph decompositions gives an 0(n3)-algorithm. Our result is less general but much simpler.


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  1. 1.
    A. BOUCHET Digraph decompositions and eulerian systems. SIAM J. Alg. Disc. Methods, vol. 8, no 3, (1987)Google Scholar
  2. 2.
    W.H. CUNNINGHAM Decomposition of directed graphs. SIAM J. Alg. Methods, vol. 3, no 2, (1982)Google Scholar
  3. 3.
    P. ERDOS, E. FRIED, A. HAJNAL and E.C. MILNER Some remarks on simple tournaments. Alg. Univ. 2, p. 238–245, (1972)Google Scholar
  4. 4.
    P. ERDOS, A. HAJNAL and E.C. MILNER Simple one-point extensions of tournaments. Mathematika, 19, p. 57–62, (1972)Google Scholar
  5. 5.
    A. MATTEO Un algorithme efficace pour le problème d'isomorphisme des graphes de Tournois. Thèse de 3e cycle, Université P. Sabatier Toulouse, (1984)Google Scholar
  6. 6.
    R.H. MOHRING, F.J. RADERMACHER Substitution decomposition for discrete structures and connection with combinatorial optimization. Annals of Discrete Mathematics 19, p. 257–356, (1984)Google Scholar
  7. 7.
    J.W. MOON Embedding Tournaments in Simple Tournaments. Discrete Math. 2, p. 389–395, (1972)CrossRefGoogle Scholar
  8. 8.
    J.W. MOON Topics on tournaments. Holt, Rinehart and Wiston, New York, (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • A. Astie-Vidal
    • 1
  • A. Matteo
    • 1
  1. 1.Laboratoire MLADUniversité Paul Sabatier ToulouseFrance

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