Full orientability of graphs with at most one dependent arc

Suppose that D is an acyclic orientation of a graph G. An arc of D is dependent if its reversal creates a directed cycle. Let () denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G. We call Gfully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying . We show that a connected graph G is fully orientable if . This generalizes the main result in Fisher et al. [D.C. Fisher, K. Fraughnaugh, L. Langley, D.B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71 (1997) 73-78].