Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] W. T. Tutte, Graph theory, Encyclopedia of Math. and Its Appl., vol. 21, Addison-Wesley, Reading, Mass., 1984. MR 746795 (87c:05001)
  • [2] Th. Zaslavsky, Biased graphs, I. Bias, balance, and gains, J. Combin. Theory Ser. B (submitted). MR 1007712 (90k:05138)
  • [3] -, Biased graphs. II. The three matroids, J. Combin. Theory Ser. B (submitted).
  • [4] -, Characterizations of signed graphs, J. Graph Theory 5 (1981), 401-406. MR 635702 (83a:05122)
  • [5] -, Signed graphs, Discrete Appl. Math. 4 (1982), 47-74. Erratum, ibid. 5 (1983), 248. MR 676405 (84e:05095a)
  • [6] -, Biased graphs whose matroids are special binary matroids (submitted).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 05C75

Retrieve articles in all journals with MSC: 05C75


Additional Information

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

American Mathematical Society