On preserving full orientability of graphs

Abstract

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 min ( G ) ( d max ( G ) ) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G . We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying d min ( G ) ⩽ k ⩽ d max ( 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 d min ( G ) .