Lovasz　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　Zaslaysky　and　generalising　the　results　of　Lovasz　and　Slilaty.　(C)　2018　Elsevier　Inc.　All　rights　reserved.