  Lov\'asz has completely characterised the structure of graphs with no two
vertex-disjoint cycles, while Slilaty has given a structural characterisation
of graphs with no two vertex-disjoint odd cycles; his result is in fact more
general, describing signed graphs with no two vertex-disjoint negative cycles.
A biased graph is a graph with a distinguished set of cycles (called balanced)
with the property that any theta subgraph does not contain exactly two balanced
cycles. In this paper we characterise the structure of biased graphs with no
two vertex-disjoint unbalanced cycles, answering a question by Zaslavsky and
generalising the results of Lov\'asz and Slilaty.
