On matrices with the Edmonds-Johnson property arising from bidirected graphs

In this paper we study totally half-modular matrices obtained from {0,±1}-matrices with at most two nonzero entries per column by multiplying by 2 some of the columns. We give an excluded-minor characterization of the matrices in this class having strong Chvàtal rank 1. Our result is a special case of a conjecture by Gerards and Schrijver [11]. It also extends a well known theorem of Edmonds and Johnson [10].