A bidirected graph is a signed graph in which the two ends of every graph edge are oriented independently. Equivalently, each half-edge 
incident with a graph vertex 
is assigned a value 
indicating a graph
orientation away from or toward 
. The two ends of a positive graph
edge have opposite orientations, while the
ends of a negative graph edge have the same graph
orientation.
For a function 
with values in an Abelian
group 
,
its boundary at 
is
 |
The function is an 
-flow if 
at every graph vertex.
A bidirected graph is flow-admissible if it has a nowhere-zero
flow over the integers. Every flow-admissible bidirected
Eulerian graph has a nowhere-zero
flow with values of magnitude less than 4 (Máčajová and Škoviera
2015), and Chen and Fan (2026) refine this by confining every graph
edge of flow magnitude 3 to one suitable circle.
