Abstract In a bidirected graph, each end of each edge is independently oriented. We show how to express any column of the incidence matrix as a half‐integral linear combination of any column basis, through a simplification, based on an idea of Bolker, of a combinatorial algorithm of Appa and Kotnyek. Corollaries are that the inverse of each nonsingular square submatrix has entries 0, $\pm{1\over 2}$ , and ±1, and that a bidirected integral linear program has half‐integral solutions. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(1), 36–38 2006

The cocycle code of an undirected graph $\Gamma $ is the linear span over ${\text{F}}_2 $ of the characteristic vectors of cutsets. (If $\Gamma $ is complete bipartite, this is the generalized Gale–Berlekamp code.) The natural bijection between the cosets of this code and the switching classes of signed graphs based on $\Gamma $ is used to show that the number of such classes is equal to the number of even-degree subgraphs of $\Gamma $ in both the labeled and unlabeled cases and to improve by coding theory previous bounds on $D( \Gamma )$, the maximum line index of imbalance of signings of $\Gamma $. Bounds on $D( \Gamma )$ are obtained in terms of the genus of $\Gamma $ and on the number of unlabeled even-degree subgraphs in terms of $D( \Gamma )$. Numerous examples are treated, including the “grid” (or “lattice”) graphs that are of interest in the Ising model of spin glasses.