Full Orientability of Graphs

Let D be an acyclic orientation of a graph G. Then an arc of D is called dependent if its reversal creates a directed cycle. Let d min ( G ) ( d max ( G ) ) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G. A graph G is called fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying d min ( G ) ⩽ d ⩽ d max ( G ) . We survey results on the fully orientability of a graph.