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# On tree width, bramble size, and expansion

@article{Grohe2009OnTW, title={On tree width, bramble size, and expansion}, author={Martin Grohe and D{\'a}niel Marx}, journal={J. Comb. Theory, Ser. B}, year={2009}, volume={99}, pages={218-228} }

- Published 2009 in J. Comb. Theory, Ser. B
DOI:10.1016/j.jctb.2008.06.004

A bramblein a graphG is a family of connected subgraphs of G such that any two of these subgraphs have a nonempty intersection or are joined by an edge. The ord r of a bramble is the least number of vertices required to cover every subgraph in the bramble. Seymour and Thomas [8] proved that the maximum order of a bramble in a graph is precisely the tree wid th of the graph plus one. We prove that every graph of tree width at least k has a bramble of order Ω(k1/2/ log2k) and size polynomial in… CONTINUE READING