Non simple tournaments : Theoretical properties and a polynomial algorithm

Abstract

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.