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

Nick Brettell, Rutger Campbell, Daryl Funk +1

We prove that a biased graph is gainable over the group ${\mathsf{Z}}_3$ if and only if it contains no minor isomorphic to $(4K_2,\varnothing )$, $\pm K_3$, or $-K_4$. We develo…