Discussiones Mathematicae Graph Theory 29(1) (2009) 51-69
doi: 10.7151/dmgt.1432

[BIBTex] [PDF] [PS]


Darren B. Parker

Department of Mathematics Grand Valley State University Allendale, MI 49401-6495, USA e-mail: parkerda@gvsu.edu

Randy F. Westhoff  and  Marty J. Wolf

Department of Mathematics & Computer Science Bemidji State University Bemidji, MN 56601, USA
e-mail: rwesthoff@bemidjistate.edu
e-mail: mjwolf@bemidjistate.edu


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.

2000 Mathematics Subject Classification: 05C20, 06B99.


[1] M. Changat and J. Mathew, On triangle path convexity in graphs, Discrete Math. 206 (1999) 91-95, doi: 10.1016/S0012-365X(98)00394-X.
[2] G. Chartrand and J.F. Fink and P. Zhang, Convexity in oriented graphs, Discrete Applied Math. 116 (2002) 115-126, doi: 10.1016/S0166-218X(00)00382-6.
[3] G. Chartrand and P. Zhang, Convex sets in graphs, Cong. Numer. 136 (1999) 19-32.
[4] P. Duchet, Convexity in graphs II. Minimal path convexity, J. Combin. Theory (B) 44 (1988) 307-316, doi: 10.1016/0095-8956(88)90039-1.
[5] P. Erdös, E. Fried, A. Hajnal and E.C. Milner, Some remarks on simple tournaments, Algebra Universalis 2 (1972) 238-245, doi: 10.1007/BF02945032.
[6] M.G. Everett and S.B. Seidman, The hull number of a graph, Discrete Math. 57 (1985) 217-223, doi: 10.1016/0012-365X(85)90174-8.
[7] D.J. Haglin and M.J. Wolf, On convex subsets in tournaments, SIAM Journal on Discrete Mathematics 9 (1996) 63-70, doi: 10.1137/S0895480193251234.
[8] F. Harary and J. Nieminen, Convexity in graphs, J. Differential Geometry 16 (1981) 185-190.
[9] R.E. Jamison and R. Nowakowski, A Helly theorem for convexity in graphs, Discrete Math. 51 (1984) 35-39, doi: 10.1016/0012-365X(84)90021-9.
[10] J.W. Moon, Embedding tournaments in simple tournaments, Discrete Math. 2 (1972) 389-395, doi: 10.1016/0012-365X(72)90016-7.
[11] J. Nieminen, On path- and geodesic-convexity in digraphs, Glasnik Matematicki 16 (1981) 193-197.
[12] D.B. Parker, R.F. Westhoff and M.J. Wolf, On two-path convexity in multipartite tournaments, European J. Combin. 29 (2008) 641-651, doi: 10.1016/j.ejc.2007.03.009.
[13] D.B. Parker, R.F. Westhoff and M.J. Wolf, Two-path convexity in clone-free regular multipartite tournaments, Australas. J. Combin. 36 (2006) 177-196.
[14] A. Abueida, W.S. Diestelkamp, S.P. Edwards and D.B. Parker, Determining properties of a multipartite tournament from its lattice of convex subsets, Australas. J. Combin. 31 (2005) 217-230.
[15] D.B. Parker, R.F. Westhoff and M.J. Wolf, Two-path convexity and bipartite tournaments of small rank, to appear in Ars Combin.
[16] J.L. Pfaltz, Convexity in directed graphs, J. Combin. Theory 10 (1971) 143-152, doi: 10.1016/0095-8956(71)90074-8.
[17] N. Polat, A Helly theorem for geodesic convexity in strongly dismantlable graphs, Discrete Math. 140 (1995) 119-127, doi: 10.1016/0012-365X(93)E0178-7.
[18] J.C. Varlet, Convexity in Tournaments, Bull. Societe Royale des Sciences de Liege 45 (1976) 570-586.
[19] M.L.J van de Vel, Theory of Convex Structures (North Holland, Amsterdam, 1993).

Received 24 September 2007
Revised 27 June 2008
Accepted 14 October 2008