Full orientability of graphs with at most one dependent arc

Hsin-Hao Lai, Ko-Wei Lih, Li-Da Tong
2009 Discrete Applied Mathematics  
Suppose that D is an acyclic orientation of a 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 d dependent arcs for every d satisfying d min (G) d d max (G). We show that a connected graph G is fully orientable if d min (G) 1. This generalizes the main result in Fisher et al.
more » ... D.C. Fisher, K. Fraughnaugh, L. Langley, D.B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71 (1997) 73-78].
doi:10.1016/j.dam.2009.04.013 fatcat:laypq2jbqfbcvgqw3rdopobouu