Tutte polynomials for oriented matroids (original) (raw)

The Tutte polynomial of some matroids

2014

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul-tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi-neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for-mulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.

A new formula for an evaluation of the Tutte polynomial of a matroid

Discrete Mathematics, 2005

Given a matroid M and its Tutte polynomial T M (x, y), T M (0, 1) is an invariant of M with various interesting combinatorial and topological interpretations. Being a Tutte-Grothendieck invariant, T M (0, 1) may be computed via deletion-contraction recursions. In this note we derive a new recursion formula for this invariant that involves contractions of M through the circuits containing a fixed element of M.

A note on some inequalities for the Tutte polynomial of a matroid

2009

We prove that if a matroid M contains two disjoint bases (or, dually, if its ground set is the union of two bases), then T M (a, a) ≤ max{T M (2a, 0), T M (0, 2a)} for a ≥ 2. This resembles the conjecture that appears in C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Annals of Combinatorics 3 (1999) pp. 417-429: If G is a 2-connected graph with no loops, then T G (1, 1) ≤ max{T G (2, 0), T (0, 2)}. We conjecture that T M (1, 1) ≤ max{T M (2, 0), T M (0, 2)} for matroids which contains two disjoint bases or its ground set is the union of two bases. We also prove the latter for some families of graphs and matroids.

The Tutte polynomial of matroids constructed by a family of splitting operations

2021

To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting off to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.

On sign-invariance graphs of uniform oriented matroids

Discrete Mathematics, 1990

Two elements of an oriented matroid constitute an invariant pair if all signed circuits containing them have the same sign (resp. different signs). The invariance graph of an oriented matroid M(E) is the graph with vertex set E and where edges are the invariant pairs. We prove that invariance graphs of uniform oriented matroids have maximum degree at most 2 (except in trivial cases) and that the alternating matroid is determined, up to reorientation, by its invariance graph. Representable uniform oriented matroids with empty invariance graphs are constructed.

On Cyclic Matroids and their Applications

arXiv (Cornell University), 2021

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such as graphs, matrices, codes and projective geometries. In this work, we define cyclic matroids as matroids over a ground set of size n whose automorphism group contains an n-cycle. We study the properties of such matroids, with special focus on the minimum size of their basis sets. For this, we broadly employ two different approaches: the multiple basis exchange property, and an orbit-stabilizer method, developed by analyzing the action of the cyclic group of order n on the set of bases. We further present some applications of our theory to algebra and geometry, presenting connections to cyclic projective planes, cyclic codes and k-normal elements.

Workshop Program Combinatorial geometries 2018 : matroids , oriented matroids and applications

2018

s of talks Monday 24 at 9h: A matroid extension result James OXLEY (Louisiana State University, USA) Let (A,B) be a 3-separation in a matroid M . If M is representable, then, in the underlying projective space, there is a line where the subspaces spanned by A and B meet, and M can be extended by adding elements from this line. In general, Geelen, Gerards, and Whittle proved that M can be extended by an independent set {p, q} such that {p, q} is in the closure of each of A and B. In this extension, each of p and q is freely placed on the line L spanned by {p, q}. This talk will discuss a result that gives necessary and su cient conditions under which a xed element can be placed on L. Monday 24 at 9h25: Su cient condition for almost irreducibility Csongor CSEHI (Budapest University of Technology and Economics, Hungary) A matroid N is almost irreducible if for any decomposition to matroid union M1 ∨M2 = N , N is a series extension of a submatroid of one of the matroids Mi ∈ {M1,M2}. An...

Matroids in terms of Cayley graphs and some related results1

The point we try to get across is that the generalization of the coun- terparts of the matroid theory in Cayley graphs since the matroid theory frequently simplify the graphs and so Cayley graphs. We will show that, for a Cayley graph G, the cutset matroid M ( G) is the dual of the circuit matroid M( G). We will also deduce that if G is an abstract-dual of a Cayley graph , then M( G) is isomorphic to (M( G)) . 2000 Mathematics Subject Classication : 05C20; 05C25; 05C60.

A note on mixed graphs and matroids

A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.