Constant Congestion Brambles

Abstract

A bramble in an undirected graph G is a family of connected subgraphs of G such that for every two subgraphs H1 and H2 in the bramble either V (H1) ∩ V (H2) ≠ θ or there is an edge of G with one endpoint in V (H1) and the second endpoint in V (H2). The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph G equals one plus the treewidth of G. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree n-vertex expander a bramble of order Ω(n1/2+δ) requires size exponential in Ω(n2δ) for any fixed δ ∈ (0, 1/2 ]. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth k admits a bramble of order Ω(k1/2) and size O(k3/2). (Ω and O hide polylogarithmic divisors and factors, respectively.) In this note, we first sharpen the second bound by proving that every graph G of treewidth at least k contains a bramble of order Ω(k1/2) and congestion 2, i.e., every vertex of G is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every δ ∈ (0, 1/2 ], every graph G of treewidth at least k contains a bramble of order Ω(k1/2+δ) and size 2O(k2δ).