On preserving full orientability of graphs

Suppose that DD is an acyclic orientation of the graph GG. An arc of DD is dependent   if its reversal creates a directed cycle. Let dmin(G)dmin(G) (dmax(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 kk dependent arcs for every kk satisfying dmin(G)⩽k⩽dmax(G)dmin(G)⩽k⩽dmax(G). In this paper, we study conditions under which full orientability of a graph can be preserved when the graph is extended by attaching new paths or cycles. Preservation theorems are applied to prove full orientability of subdivisions of Halin graphs and graphs of maximum degree at most three. We also characterize their dmin(G)dmin(G).