Graph searching and a min-max theorem for tree-width
Journal of Combinatorial Theory Series B
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs
European Journal of Combinatorics
Tractable hypergraph properties for constraint satisfaction and conjunctive queries
Proceedings of the forty-second ACM symposium on Theory of computing
On brambles, grid-like minors, and parameterized intractability of monadic second-order logic
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Treewidth of Erdős-Rényi random graphs, random intersection graphs, and scale-free random graphs
Discrete Applied Mathematics
Large-treewidth graph decompositions and applications
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries
Journal of the ACM (JACM)
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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 @W(k^1^/^2/log^2k) and size polynomial in n and k, and that for every k there is a graph G of tree width @W(k) such that every bramble of G of order k^1^/^2^+^@e 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.