A characterization of unimodular orientations of simple graphs

Let G be a simple graph, and consider an orientation of the edges of G. Where V is the vertex-set of G, let A = (A,, : u, w E V) be the integral antisymmetric (0, +_ 1)-matrix such that A,,, = 1 if and only if VW is an edge of G directed from u to w. The orientation of G is said to be unimodular if for every WC V the determinant of the submatrix (A,, : u, w E W) is 0 or 1. For example, it is known that circle graphs can be provided with unimodular orientations. For any vertex v and any subset of vertices W let E(D, W) be the number of edges leaving u and entering W less the number of edges entering u and leaving IV. The main result of this paper says that the orientation of G is unimodular if and only if for every subset of vertices W such that E(W, W) is even for every w E W, we can reverse the orientations of the edges which belong to some cocycle C so that (E(u, W)( < 1 becomes true for every vertex. This generalizes a result of Ghouila-Houri for totally unimodular matrices.