The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives

Vergnas, Michel Las

doi:10.5802/aif.1702

Cited by 26 publications

(24 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Whitney planarity criteria [12] says that a graph G is planar if and only if its bond matroid B.G/ is the cycle matroid of some graph. In this case, it will be the cycle matroid of the dual graph, [7], and [8]). For two matroids M and…”

Section: Matroids and The Las Vergnas Polynomialmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial invariants of graphs on surfaces

Askanazi¹,

Chmutov²,

Estill³

et al. 2012

Quantum Topol.

View full text Add to dashboard Cite

Abstract. For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This gives an expression of the polynomial, defined by M. Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in the first homology group of the surface.Mathematics Subject Classification (2010). 05C10, 05C31, 57M15, 57M25, 57M27.

show abstract

Section: Matroids and The Las Vergnas Polynomialmentioning

confidence: 99%

“…Various combinatorial parameters of the matroids C.G/ and B.G / can be assembled into the Las Vergnas polynomial of a matroid perspective B.G / ! C.G/ introduced in [6], [7], and [8].…”

Section: Introductionmentioning

confidence: 99%

Polynomial invariants of graphs on surfaces

Askanazi¹,

Chmutov²,

Estill³

et al. 2012

Quantum Topol.

View full text Add to dashboard Cite

show abstract

“…Las Vergnas (Theorem 5.3 of [16]) showed that T M →M ′ satisfies deletion-contraction relations that provide a complete recursive definition of the polynomial. (1) if e ∈ E is neither an isthmus nor a loop of M , then…”

Section: 2mentioning

confidence: 99%

“…The Las Vergnas polynomial was first defined in terms of the combinatorial geometry of an embedded graph (i.e., via circuit matroids). It arises as a special case in his much larger body of work on the Tutte polynomial of a morphism of matroids (see [9,15,16,17,18,19]). We present here a discussion of matroid perspectives in the special context of embedded graph theory.…”

Section: Introductionmentioning

confidence: 99%

The Las Vergnas polynomial for embedded graphs

Ellis-Monaghan

Moffatt

2015

European Journal of Combinatorics

View full text Add to dashboard Cite

This paper is dedicated to the memory of Michel Las Vergnas in gratitude for not only so much beautiful mathematics, but also many instances of very kind and insightful correspondence.Abstract. The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobás-Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas' results on Eulerian circuits from graphs in surfaces of low genus to surfaces of arbitrary genus.

show abstract

“…We have introduced in 1975 [12] the 3-variable Tutte polynomial of a matroid perspective (definition in Section 2), and studied its properties in a series of papers: fundamental properties in [13] [15], Eulerian partitions of 4-valent graphs imbedded in surfaces [14], activities of orientations in [16] [18], vectorial matroids in [8], computational complexity in [17]. The Tutte polynomial of a matroid perspective may equivalently be considered as associated with a ported matroid, i.e.…”

mentioning

confidence: 99%