Bounding and stabilizing realizations of biased graphs with a fixed group

Abstract

Given a group Γ and a biased graph (G,B), we define a what is meant by a Γ-realization of (G,B) and a notion of equivalence of Γ-realizations. We prove that for a finite group Γ and t≥3, there are numbers n(Γ) and n(Γ,t) such that the number of Γ-realizations of a vertically 3-connected biased graph is at most n(Γ) and that the number of Γ-realizations of a nonseparable biased graph without a (2Ct,∅)-minor is at most n(Γ,t). Other results pertaining to contrabalanced biased graphs are presented as well as an analogue to Whittle's Stabilizer Theorem for Γ-realizations of biased graphs.