The excluded minors for ${\mathsf{Z}}_3$-gainable and regular biased graphs

We prove that a biased graph is gainable over the group ${\mathsf{Z}}_3$ if and only if it contains no minor isomorphic to $(4K_2,\mathrm{\varnothing })$, $\pm K_3$, or $-K_4$. We develop a theory of “partial groups” that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over ${\mathsf{Z}}_2$ and ${\mathsf{Z}}_3$. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to $(3K_2,\mathrm{\varnothing })$, $\pm K_3$, or $-K_4$.