Chordal Graphs are Fully Orientable

Abstract

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 \(d_{\min}(G)\) (\(d_{\max}(G)\)) 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 \(d_{\min}(G) \leq d \leq d_{\max}(G)\). 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.