Entanglements

Robertson and Seymour constructed for every graph G a tree-decomposition that efficiently distinguishes all the tangles in G . While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical. The key ingredient is an axiomatisation of ‘local properties’ of tangles. Generalisations to locally finite graphs and matroids are also discussed.