Orientable convexity, geodetic and hull numbers in graphs
We prove three results conjectured or stated by Chartrand and Zhang [European J. Combin. 21 (2000) 181189] and Chartrand et al. [Discrete Appl. Math. 116 (2002) 115126; Internat. J. Math. Math. Sci. 36 (2003) 22652275]: a connected graph has orientations with different geodetic numbers, orientations with different hull numbers, and, if there are no end-vertices, orientations with different convexity numbers. The proof of the first result is a correction of Chartrand and Zhang's proof, and allows for an easy proof of the second result. The third result says roughly that graphs without end-vertices can be oriented anti-transitively.
convex hull number oriented graph geodesic transitively orientable