Biased graphs with no two vertex-disjoint unbalanced cycles

Lovász 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 Lovász and Slilaty.