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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1432,
author = {Darren B. Parker and Randy F. Westhoff and Marty J. Wolf},
title = {Convex independence and the structure of clone-free multipartite tournaments},
journal = {Discussiones Mathematicae Graph Theory},
volume = {29},
year = {2009},
pages = {51-69},
zbl = {1213.05117},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1432}
}
Darren B. Parker; Randy F. Westhoff; Marty J. Wolf. Convex independence and the structure of clone-free multipartite tournaments. Discussiones Mathematicae Graph Theory, Tome 29 (2009) pp. 51-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1432/
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