Graphs and Combinatorics

, Volume 18, Issue 2, pp 209–217

The Convexity Number of a Graph

Authors

  • Gary Chartrand
    • Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA.
  • Curtiss E. Wall
    • Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA
  • Ping Zhang
    • Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA. e-mail: zhang@math-stat.wmich.edu

DOI: 10.1007/s003730200014

Cite this article as:
Chartrand, G., Wall, C. & Zhang, P. Graphs Comb (2002) 18: 209. doi:10.1007/s003730200014

Abstract.

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a uv shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,vS. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤lkn−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented.

Key words. Convex set, Convexity number

Copyright information

© Springer-Verlag Tokyo 2002