780 research outputs found
Tangles, tree-decompositions, and grids in matroids
A tangle in a matroid is an obstruction to small branch-width. In particular, the maximum order of a tangle is equal to the branch-width. We prove that: (i) there is a tree-decomposition of a matroid that ādisplaysā all of the maximal tangles, and (ii) when M is representable over a ļ¬nite ļ¬eld, each tangle of suļ¬ciently large order ādominatesā a large grid-minor. This extends results of Robertson and Seymour concerning Graph Minors
Branch-width and well-quasi-ordering in matroids and graphs
AbstractWe 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 branch-width 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
Quasi-graphic matroids
Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a rame matroid
On inequivalent representations of matroids over non-prime ļ¬elds
For each ļ¬nite ļ¬eld of prime order there is a constant such that every 4-connected matroid has at most inequivalent representations over . We had hoped that this would extend to all ļ¬nite ļ¬elds, however, it was not to be. The -mace is the matroid obtained by adding a point freely to . For all , the -mace is 4-connected and has at least representations over any ļ¬eld of non-prime order . More generally, for , the -mace is vertically -connected and has at least inequivalent representations over any ļ¬nite ļ¬eld of non-prime order
REACH implementation costs in the Belgian food industry:a semi-qualitative study
In this paper we discuss how companies in the Belgian food industry are affected by the REACH legislation and whether their competitiveness is weakened as a result. The study has been carried out through an extensive literature study, an electronic survey, in-depth interviews and a case-study. No indication is observed of REACH compliance significantly hampering the competitive position of Belgian food industry. The overall cost burden seems to be relatively low. In contrast with the chemical industry, large food companies bear the highest costs, whereas the financial impact on small and medium-sized food companies remains limited.<br
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