, Volume 18, Issue 2, pp 209-217

The Convexity Number of a Graph

Purchase on Springer.com

$39.95 / €34.95 / £29.95*

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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.

Received: August 19, 1998 Final version received: May 17, 2000