Convexity in oriented graphs

For vertices u and nu in an oriented graph D, the closed interval I[u, nu] consists of u and nu together with all vertices lying in a u - nu geodesic or nu - u geodesic in D. For S subset of or equal to V(D), I[S] is the union of all closed intervals I[u, nu] with u, nu epsilon S. A set S is convex if I[S] = S. The convexity number con(D) is the maximum cardinality of a proper convex set of V(D). The nontrivial connected oriented graphs of order n with convexity number n - 1 are characterized. It is shown that there is no connected oriented graph of order at least 4 with convexity number 2 and that every pair k, n of integers with 1 less than or equal to k less than or equal to n - 1 and k not equal 2 is realizable as the convexity number and order, respectively, of some connected oriented graph. For a nontrivial connected graph G, the lower orientable convexity number con(-)(G) is the minimum convexity number among all orientations of G and the upper orientable convexity number con(+)(G) is the maximum such convexity number. It is shown that con(+) (G) = n - 1 for every graph G of order n greater than or equal to 2. The lower orientable convexity numbers of some well-known graphs are determined, with special attention given to outerplanar graphs. (C) 2002 Elsevier Science B.V. All rights reserved.