Convex independence and the structure of clone-free multipartite tournaments
Darren B. Parker, Randy F. Westhoff, Marty J. Wolf · Discussiones Mathematicae Graph Theory · 2009
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.