  A graphic arrangement is a subarrangement of the braid arrangement whose set
of hyperplanes is determined by an undirected graph. A classical result due to
Stanley, Edelman and Reiner states that a graphic arrangement is free if and
only if the corresponding graph is chordal, i.e., the graph has no chordless
cycle with four or more vertices. In this article we extend this result by
proving that the module of logarithmic derivations of a graphic arrangement has
projective dimension at most one if and only if the corresponding graph is
weakly chordal, i.e., the graph and its complement have no chordless cycle with
five or more vertices.
