Simplicial Decompositions, Tree-decompositions and Graph Minors
نویسنده
چکیده
The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used the following idea. He considered all (edge-maximal finite) graphs not subcontracting to K5, and proved that breaking up any such graph along separating complete subgraphs (‘simplices’) leaves factors that are either planar or isomorphic to a certain 3-chromatic non-planar graph. Assuming the 4CC, these factors are therefore 4-colourable, a property which can be lifted back to the original graph. Wagner’s decompositions were later redefined—and named ‘simplicial decompositions’—by Halin [ 9 ], to make them suitable for infinite graphs; the definition given by Halin is equivalent to our conditions (S1)–(S3). It is an interesting fact that for finite graphs the conditions (S1)–(S3) imply (S4), which is not the case for infinite graphs. Thus, with the transition to infinite graphs based on (S1)–(S3), one of the most striking features of Wagner’s finite decompositions was lost: their ‘tree shape’, a consequence of (S4) (see [ 2 ] for details). It was this ‘tree shape’ that gave rise to the other generalization of Wagner’s decompositions: the ‘tree-decompositions’ recently introduced by Robertson and Seymour [ 12 ]. Robertson and Seymour’s definition of a tree-decomposition (again for finite graphs) is equivalent to our conditions (S1), (S3) and (S4).
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