The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski [8] in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial, and a universal group, called the Tutte-Grothend...

We define and study semimatroids, a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We show that geometric semilattices are precisely the posets of flats of semimatroids. We define and investigate the Tutte polynomial of a semimatroid. We prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial inte...

We provide a new proof of Brylawski’s formula for the Tutte polynomial of the tensor product of two matroids. Our proof involves extending Tutte’s formula, expressing the Tutte polynomial using a calculus of activities, to all polynomials involved in Brylawski’s formula. The approach presented here may be used to show a signed generalization of Brylawski’s formula, which may be used to compute ...

We investigate the Tutte polynomial f(P; t, z) of a series-parallel partially ordered set P. We show that f(P) can be computed in polynomial-time when P is series-parallel and that series-parallel posets having isomorphic deletions and contractions are themselves isomorphic. A formula forf’(P*) in terms off(P) is obtained and shows these two polynomials factor over Z[t, z] the same way. We exam...

Negami and Kawagoe has already defined a polynomial f(G) associated with each graph G as what discriminates graphs more finely than the polynomial f(G) definedby Negami and the Tutte polynomial. In this paper, we shall show that the polynomial f(G) includes potentially the generating function counting the independent sets and the degree sequence of a graph G, which cannot be recognized from f(G...

Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this article, we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove in this framework that every recursive definition of a graph po...

Matroid duality is an important generalization of duality for planar graphs. Using unpublished notes of Brylawski, we extend this notion to arbitrary set systems. This allows one to define a generalized Tutte polynomial. We examine this polynomial for several set systems that are not matroids, and we also investigate the combinatorial significance of duality for these set systems.