Tutte defined a k-separation of a matroid M to be a partition ðA; BÞ of the ground set of M such that jAj; jBjXk and rðAÞ þ rðBÞ À rðMÞok: If, for all mon; the matroid M has no mseparations, then M is n-connected. Earlier, Whitney showed that ðA; BÞ is a 1-separation of M if and only if A is a union of 2-connected components of M: When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M: r

We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branchwidth are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off. © 2002 Elsevier Science (USA)

The parameterized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored. Concerning codes, the main results are that Maximum-Likelihood Decoding and Weight Distribution are hard for the parametrized complexity class W 1]. The NP-completeness of these two problems was established by Berlekamp, McEliece, and van Tilborg in 1978, using a reduction from 3-Dimensional Matching. On the other hand, our proof of hardness for W 1] is based on a parametric polynomialtime transformation from Perfect Code in graphs. An immediate consequence of our results is that bounded-distance decoding is likely to be hard for linear codes. Concerning lattices, we address the Theta Series problem of determining, for an integer lattice and a positive integer k, whether there is a vector x 2 of Euclidean norm k. We prove here for the rst time that Theta Series is NP-complete, and show that it is also hard for W 1]. We furthermore prove that the Nearest Vector problem for integer lattices is hard for W 1]. These problems are the counterparts of Weight Distribution and Maximum-Likelihood Decoding for lattices. Relations between all these problems and combinatorial problems in graphs are discussed.

Abstract. For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.

A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a, b g P, a q b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of minors, duals, direct sums, and 2-sums. Homomorphisms of partial fields are defined. It is shown that if : P ª P is a non-trivial partial-field homomorphism, 1 2 then every matroid representable over P is representable over P . The connec-1 2 tion with Dowling group geometries is examined. It is shown that if G is a finite abelian group, and r ) 2, then there exists a partial field over which the rank-r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots. ᮊ