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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Vertices of localized imbalance in a biased graph


Author: Thomas Zaslavsky
Journal: Proc. Amer. Math. Soc. 101 (1987), 199-204
MSC: Primary 05C75
DOI: https://doi.org/10.1090/S0002-9939-1987-0897095-1
MathSciNet review: 897095
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Abstract: A biased graph consists of a graph $ \Gamma $ and a subclass $ \mathcal{B}$ of the polygons of $ \Gamma $, such that no theta subgraph of $ \Gamma $ contains exactly two members of $ \mathcal{B}$. A subgraph is balanced when all its polygons belong to $ \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.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0897095-1
Keywords: Biased graph, balancing vertex, signed graph
Article copyright: © Copyright 1987 American Mathematical Society