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Book Sections Year : 2022

The Tutte polynomial of oriented matroids

Emeric Gioan

Abstract

This chapter covers properties of the Tutte polynomial for oriented matroids. It covers the basics of oriented matroids, as well as their interactions with the Tutte polynomial. Oriented matroids, interpretations and translations in hyperplane arrangements and in directed graphs. The Tutte polynomial in terms of orientation-activities, generalization to oriented-matroid perspectives, and a 4-variable expansion. Geometric interpretations of the β-invariant, of the other coefficients, and of particular evaluations. Expression of the Tutte polynomial of a matroid in terms of active filtrations/partitions and β-invariants of minors. Activity-preserving bijections between bases/subsets/no-broken-circuit-subsets and activity-classes/reorientations/regions. Circuit/cocircuit reversal classes in directed graphs and regular matroids.
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Dates and versions

lirmm-03868723 , version 1 (23-11-2022)

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Emeric Gioan. The Tutte polynomial of oriented matroids. Handbook of the Tutte Polynomial and Related Topics, Chapter 31, Chapman and Hall/CRC, 25 p., 2022, ⟨10.1201/9780429161612⟩. ⟨lirmm-03868723⟩
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