On fully orientability of 2-degenerate graphs

Suppose that D is an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define dmin(G) (dmax(G)) to be the minimum (maximum) number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying dmin(G)⩽k⩽dmax(G). We prove that every 2-degenerate graph is fully orientable and give interpretations to their dmin.