Non simple tournaments : Theoretical properties and a polynomial algorithm
The first part of this paper is a theoretical study of the properties of the convex subsets of a tournament.
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  about the decomposition of directed graphs, applied to regular tournaments allow to determine their convex subsets in time 0(n4).
Very recent results of  about digraph decompositions gives an 0(n3)-algorithm. Our result is less general but much simpler.
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