Graphs and Combinatorics

, Volume 18, Issue 2, pp 209–217

The Convexity Number of a Graph

  • Gary Chartrand
  • Curtiss E. Wall
  • Ping Zhang

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

Authors and Affiliations

  • Gary Chartrand
    • 1
  • Curtiss E. Wall
    • 2
  • Ping Zhang
    • 3
  1. 1.Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA.US
  2. 2.Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USAUS
  3. 3.Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA. e-mail: zhang@math-stat.wmich.eduUS