Orientable convexity, geodetic and hull numbers in graphs

We prove three results conjectured or stated by Chartrand, Fink and Zhang [European J. Combin 21 (2000) 181–189, Disc. Appl. Math. 116 (2002) 115–126, and pre-print of “The hull number of an oriented graph”]. For a digraph D, Chartrand et al. defined the geodetic, hull and convexity number — g(D), h(D) and con(D), respectively. For an undirected graph G, g(G) and g(G) are the minimum and maximum geodetic numbers over all orientations of G, and similarly for h(G), h(G), con(G) and con(G). Chartrand and Zhang gave a proof that g(G) < g(G) for any connected graph with at least three vertices. We plug a gap in their proof, allowing us also to establish their conjecture that h(G) < h(G). If v is an end-vertex, then in any orientation of G, v is either a source or a sink. It is easy to see that graphs without end-vertices can be oriented to have no source or sink; we show that, in fact, we can avoid all extreme vertices. This proves another conjecture of Chartrand et al., that con(G) < con(G) iff G has no end-vertices.