Vertices of localized imbalance in a biased graph
Thomas Zasĺavsky · Proceedings of the American Mathematical Society · 1987
A biased graph consists of a graph Γ \Gamma and a subclass B \mathcal {B} of the polygons of Γ \Gamma , such that no theta subgraph of Γ \Gamma contains exactly two members of B \mathcal {B} . A subgraph is balanced when all its polygons belong to B \mathcal {B} . A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph having more than one balancing vertex is an unbalanced series or parallel connection of balanced graphs.