Orientation of Signed Graphs

A graph with signed arcs is oriented by directing each end of each arc in accordance with a sign-compatibility rule. We prove that the regions of the hyperplane representation of a signed graph ∑, as well as the vertices of the convex hull of all degree vectors of orientations of ∑, are in natural one-to-one correspondence with the cyclic orientations of ∑ The proof uses the oriented matroid of a signed graph. For use elsewhere, we also develop the relationships between orientations and hyperplane representations of a signed graph and those of its double covering graph