# Convex independence and the structure of clone-free multipartite tournaments

• Volume: 29, Issue: 1, page 51-69
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Darren B. Parker, Randy F. Westhoff, and Marty J. Wolf. "Convex independence and the structure of clone-free multipartite tournaments." Discussiones Mathematicae Graph Theory 29.1 (2009): 51-69. <http://eudml.org/doc/270650>.

@article{DarrenB2009,
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.},
author = {Darren B. Parker, Randy F. Westhoff, Marty J. Wolf},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {convex sets; rank; Helly number; Radon number; multipartite tournaments},
language = {eng},
number = {1},
pages = {51-69},
title = {Convex independence and the structure of clone-free multipartite tournaments},
url = {http://eudml.org/doc/270650},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Darren B. Parker
AU - Randy F. Westhoff
AU - Marty J. Wolf
TI - Convex independence and the structure of clone-free multipartite tournaments
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 1
SP - 51
EP - 69
AB - 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.
LA - eng
KW - convex sets; rank; Helly number; Radon number; multipartite tournaments
UR - http://eudml.org/doc/270650
ER -

## References

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