Geodesic
An extension of matroid rank submodularity and the \(Z\)-Rayleigh property
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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We define an extension of matroid rank submodularity called $R$-submodularity, and introduce a minor-closed class of matroids called extended submodular matroids that are well-behaved with respect to $R$-submodularity. We apply $R$-submodularity to study a class of matroids with negatively correlated multivariate Tutte polynomials called the $Z$-Rayleigh matroids. First, we show that the class of extended submodular matroids are $Z$-Rayleigh. Second, we characterize a minor-minimal non-$Z$-Rayleigh matroid using its $R$-submodular properties. Lastly, we use $R$-submodularity to show that the Fano and non-Fano matroids (neither of which is extended submodular) are $Z$-Rayleigh, thus giving the first known examples of $Z$-Rayleigh matroids without the half-plane property.
DOI : 10.37236/600
Classification : 05B35
Mots-clés : \(R\)-submodularity, extended submodular matroids 
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     title = {An extension of matroid rank submodularity and the {\(Z\)-Rayleigh} property},
     journal = {The electronic journal of combinatorics},
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Arun P. Mani. An extension of matroid rank submodularity and the \(Z\)-Rayleigh property. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/600

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