Full orientability of graphs with at most one dependent arc

Suppose that DD is an acyclic orientation of a graph GG. An arc of DD is dependent   if its reversal creates a directed cycle. Let dmin(G) (dmax(G)) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of GG. We call GGfully orientable   if GG has an acyclic orientation with exactly dd dependent arcs for every dd satisfying dmin(G)⩽d⩽dmax(G). We show that a connected graph GG is fully orientable if dmin(G)⩽1. 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].