ArticleOriginal scientific text
Title
Convex independence and the structure of clone-free multipartite tournaments
Authors 1, 2, 2
Affiliations
- Department of Mathematics, Grand Valley State University, Allendale, MI 49401-6495, USA
- Department of Mathematics & Computer Science, Bemidji State University, Bemidji, MN 56601, USA
Abstract
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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Pages:
51-69
Main language of publication
English
Received
2007-09-24
Accepted
2008-06-27
Published
2009
DOI: 10.7151/dmgt.1432
Exact and natural sciences