Note that if a signed Eulerian graph contains an odd number of negative edges, it must contain an unbalanced cycle.
0] denote a signed Eulerian graph formed by the union of ([x.
b] and C', is an unbalanced Eulerian graph, so it contains an unbalanced cycle.
a], which is the symmetric difference of Cl and the symmetric difference of C' and Cb, is an unbalanced Eulerian graph, so it contains an unbalanced cycle.
l] and C is an unbalanced Eulerian graph that contains an unbalanced cycle of length shorter than the length of the critical cycle [C.
Polynomial-time algorithms are known for switching to a triangle-free graph [Hay96, HHW02], to a claw-free graph [JK08], to an eulerian graph
[HHW02], to a bipartite graph [HHW02], and to a planar graph [EHHR00a, Kra03].