The convexity spectra of graphs

Let D be a connected oriented graph. A set S subset of V (D) is convex in D if, for every pair of vertices x, y epsilon S, the vertex set of every x - y geodesic (x - y shortest dipath) and y - x geodesic in D is contained in S. The convexity number con(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. Let G be a graph. We define that S-C(G) = {con(D): D is an orientation of G} and S-SC(G) = {con(D): D is a strongly connected orientation of G}. In the paper, we show that, for any n >= 4, 1 <= a <= n - 2, and a not equal 2, there exists a 2-connected graph G with it vertices such that S-C(G) = S-SC(G) = {a, n - 1} and there is no connected graph G of order n >= 3 with S-SC(G) = {n - 1}. Then, we determine that S-C(K-3) = {1, 2}, S-C(K-4) = {1, 3}, S-SC(K-3)=S-SC(K-4)={1}, S-C(K-5)={1, 3, 4}, S-C(K-6)={1, 3, 4, 5}, S-SC(K-5)=S-SC(K-6)={1, 3}, S-C(K-n)={1, 3, 5, 6, ... n-1}, S-SC(K-n) = {1, 3, 5, 6, ... n - 2} for n >= 7. Finally, we prove that, for any integers n, m, and k with n >= 5, n + 1 <= m <= (2(n)) - 1 <= k <= n - 1, and k not equal 2, 4, there exists a strongly connected oriented graph D with n vertices, in edges, and convexity number k. (C) 2007 Published by Elsevier B.V.