# The Convexity Number of a Graph

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 *u*−*v* 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*,*v*∈*S*. 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≤*l*≤*k*≤*n*−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.