  We define dual-critical graphs as graphs having an acyclic orientation, where
the indegrees are odd except for the unique source. We have very limited
knowledge about the complexity of dual-criticality testing. By the definition
the problem is in NP, and a result of Bal\'azs and Christian Szegedy provides a
randomized polynomial algorithm, which relies on formal matrix rank computing.
It is unknown whether dual-criticality test can be done in deterministic
polynomial time. Moreover, the question of being in co-NP is also open.
  We give equivalent descriptions for dual-critical graphs in the general case,
and further equivalent descriptions in the special cases of planar graphs and
3-regular graphs. These descriptions provide polynomial algorithms for these
special classes. We also give an FPT algorithm for a relaxed version of
dual-criticality called $k$-dual-criticality.
