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No abstract

Adapting the method introduced in Graph Minors X [6], we propose a new proof of the duality between the bramble-number of a graph and its tree-width. This proof is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The technique simplifies the proof of bramble/tree-width duality since it does not rely on Menger's theorem. One can also derive from it all known dual notions of other classical width-parameters. Finally, it provides a dual for matroid tree-width.