Transitive closure and transitive reduction in bidirected graphs
  In a bidirected graph an edge has a direction at each end, so bidirected graphs generalize directed graphs. We generalize the definitions of transitive closure and transitive reduction from directed graphs to bidirected graphs by introducing new notions of bipath and bicircuit that generalize directed paths and cycles. We show how transitive reduction is related to transitive closure and to the matroids of the signed graph corresponding to the bidirected graph. 
Ouahiba Bessouf and Abdelkader Khelladi and Thomas Zaslavsky
math.CO