Regular Article
On Circuit Valuation of Matroids

https://doi.org/10.1006/aama.2000.0716Get rights and content
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Abstract

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R  {−∞})-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R  {−∞})-valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

Keywords

valuated matroids
bases
circuits
duality

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This work is supported by a grant-in-aid from the Ministry of Education, Science, Sports and Culture of Japan.