Rayleigh Matroids

  • Authors:

  • Youngbin Choe;David G. Wagner

  • Affiliations:

  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (e-mail: ybchoe@math.uwaterloo.ca, dgwagner@math.uwaterloo.ca);Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (e-mail: ybchoe@math.uwaterloo.ca, dgwagner@math.uwaterloo.ca)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2006

  • Balanced matroids

    STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing

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Abstract

Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail [9] in 1992. We prove a variety of results relating Rayleigh matroids to other well-known classes – in particular, we show that a binary matroid is Rayleigh if and only if it does not contain $\mathcal{S}_{8}$ as a minor. This has the consequence that a binary matroid is balanced if and only if it is Rayleigh, and provides the first complete proof in print that $\mathcal{S}_{8}$ is the only minor-minimal binary non-balanced matroid, as claimed in [9]. We also give an example of a balanced matroid which is not Rayleigh.