Geodesic
Convex independence and the structure of clone-free multipartite tournaments
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 51-69

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We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specifically, we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given convexly independent set. We use this to prove that the Helly number, Radon number, and rank coincide in any clone-free bipartite tournament. We then study the relationship between Helly independence and Radon independence in clone-free multipartite tournaments. We show that if the rank is at least 4 or the Helly number is at least 3, then the Helly number and the Radon number are equal.
Keywords: convex sets, rank, Helly number, Radon number, multipartite tournaments 
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Parker, Darren; Westhoff, Randy; Wolf, Marty. Convex independence and the structure of clone-free multipartite tournaments. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 51-69. http://geodesic.mathdoc.fr/item/DMGT_2009_29_1_a3/