On tree width, bramble size, and expansion

A bramble in a graph G 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 order of a bramble is the least number of vertices required to cover every subgraph in the bramble. Seymour and Thomas [P.D. Seymour, R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993) 22–33] proved that the maximum order of a bramble in a graph is precisely the tree width of the graph plus one. We prove that every graph of tree width at least k has a bramble of order Ω ( k 1 / 2 / log 2 k ) and size polynomial in n and k, and that for every k there is a graph G of tree width Ω ( k ) such that every bramble of G of order k 1 / 2 + ϵ has size exponential in n. To prove the lower bound, we establish a close connection between linear tree width and vertex expansion. For the upper bound, we use the connections between tree width, separators, and concurrent flows.