Vertices of localized imbalance in a biased graph

Zaslavsky, Thomas

doi:10.1090/S0002-9939-1987-0897095-1

Vertices of localized imbalance in a biased graph
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by Thomas Zaslavsky

Proc. Amer. Math. Soc. 101 (1987), 199-204

DOI: https://doi.org/10.1090/S0002-9939-1987-0897095-1

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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.

References

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Thomas Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989), no. 1, 32–52. MR 1007712, DOI 10.1016/0095-8956(89)90063-4

Biased graphs

The three matroids

Thomas Zaslavsky, Characterizations of signed graphs, J. Graph Theory 5 (1981), no. 4, 401–406. MR 635702, DOI 10.1002/jgt.3190050409
Thomas Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47–74. MR 676405, DOI 10.1016/0166-218X(82)90033-6

Biased graphs whose matroids are special binary matroids