On signed digraphs with all cycles negative

It is known that signed graphs with all cycles negative are those in which each block is a negative cycle or a single line. We now study the more difficult problem for signed digraphs. In particular we investigate the structure of those digraphs whose arcs can be signed (positive or negative) so that every (directed) cycle is negative. Such digraphs are important because they are associated with qualitatively nonsingular matrices. We identify certain families of such digraphs and characterize those symmetric digraphs which can be signed so that every cycle is negative.