The Tutte polynomial of a ported matroid
type: Item
instance of: scholarly article
title: The Tutte polynomial of a ported matroid
zbMATH Open document ID: 0614.05017
DOI: 10.1016/0095-8956(89)90010-5
author: Seth Chaiken
published in: Journal of Combinatorial Theory. Series B
publication date: 1989-01-01T00:00:00Z
review text: Las Vergnas' generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariants f is modified so the Tutte decomposition \(f(M)=f(M\setminus e)+f(M/e)\) is applied only when \(e\not\in P\) (and e is neither a loop nor an isthmus) where P is a distinguished set of points called ports. The resulting ''P- ported'' Tutte polynomial \(t_ P\) has variables \(z,w;q_ 1,q_ 2,...,q_ m\); the q's are connected matroids on subsets of P. We express \(t_ P\) of the P-ported matroid sum and cosum of \(M_ 1\) and \(M_ 2\), only taken when \(M_ 1\cap M_ 2\subseteq P\), in terms of \(t_ P(M_ 1)\) and \(t_ P(M_ 2)\). The behavior of \(t_ P(M)\) under contraction/deletion of points in P is given. Our development is based on a P-ported rank generating function. The major results are generalizations of work of Brylawski on series and parallel connections. Relationships to the geometric lattice of M, to Las Vergnas' Tutte polynomial of a matroid pointed by a family of sets, and to electrical network theory are given.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 3993603
zbMATH Keywords: combinatorial geometry
zbMATH Keywords: matroid sum
zbMATH Keywords: matroid union
zbMATH Keywords: Tutte polynomial
zbMATH Keywords: matroids
MaRDI profile type: Publication
full work available at URL: 0095-8956(89)90010-5
OpenAlex ID: W2066992144
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description: scientific article; zbMATH DE number 3993603