Chordal Graphs are Fully Orientable

TL;DR

This paper proves that all chordal graphs can be oriented acyclically to have any number of dependent arcs between the minimum and maximum, demonstrating their full orientability.

Contribution

It establishes that all chordal graphs are fully orientable, a new property linking graph structure to orientation flexibility.

Findings

All chordal graphs are fully orientable.

Chordal graphs can realize all intermediate numbers of dependent arcs.

The result connects chordal structure with orientation properties.

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 m and M denote the minimum and the 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 m <= d <= M. 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.