Chordal Graphs are Fully Orientable

Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let dmin(G) (dmax(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 dmin(G) 6 d 6 dmax(G). A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable. Keyword: acyclic orientation; full orientability; simplicial vertex; chordal graph. ∗The corresponding author