The convexity spectra of graphs

Let D be a connected oriented graph. A set S ⊆ V (D) is convex in D if, for every pair of vertices x, y ∈ 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 and SC(G) = {con(D) : D is an orientation of G} and SSC(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 6= 2, there exists a 2connected graph G with n vertices such that SC(G) = SSC(G) = {a, n−1} and there is not any connected graph G of order n ≥ 3 with SSC(G) = {n−1}. Then, we determine that SC(K3) = {1, 2}, SC(K4) = {1, 3}, SSC(K3) = SSC(K4) = {1}, SC(K5) = {1, 3, 4}, SC(K6) = {1, 3, 4, 5}, SSC(K5) = SSC(K6) = {1, 3}, SC(Kn) = {1, 3, 5, 6, ..., n − 1}, SSC(Kn) = {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 ≤ (n 2 ) − 1, 1 ≤ k ≤ n− 1, and k 6= 2, 4, there exists a strongly connected oriented graph D with n vertices, m edges, and convexity k. ∗This research was partially supported by the National Science Council under grant NSC95-2115M-110-012-MY2. National Center of Theoretical Sciences. [email protected] †1991 Mathematics Subject Classification. 05C20, 05C35, 05C15