On an interpolation property of outerplanar graphs

Let D be an acyclic orientation of a graph G. An arc of D is said to be dependent   if its reversal creates a directed cycle. Let d(D)d(D) denote the number of dependent arcs in D  . Define dmin(G)dmin(G) (dmax(G)dmax(G)) to be the minimum (maximum) number of d(D)d(D) over all acyclic orientations D of G  . We determine dmin(G)dmin(G) for an outerplanar graph G and prove that G has an acyclic orientation with exactly k   dependent arcs if dmin(G)⩽k⩽dmax(G)dmin(G)⩽k⩽dmax(G).