The convexity number of a graph

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 E 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 w(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 greater than or equal to 3 and 2 less than or equal to w(G) less than or equal to con(G) less than or equal to n - 1. It is shown that for every triple l, k, n of integers with n greater than or equal to 3 and 2 less than or equal to l less than or equal to k less than or equal to n - 1, there exists a noncomplete connected graph G of order n with w(G) = l and con(G) = k. Other results on convex numbers are also presented.