Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Easy problems for tree-decomposable graphs
Journal of Algorithms
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Finding approximate separators and computing tree width quickly
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Finding Minimal Forbidden Minors Using a Finite Congruence
ICALP '91 Proceedings of the 18th International Colloquium on Automata, Languages and Programming
Better Algorithms for the Pathwidth and Treewidth of Graphs
ICALP '91 Proceedings of the 18th International Colloquium on Automata, Languages and Programming
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs
Journal of the ACM (JACM)
Rank-width is less than or equal to branch-width
Journal of Graph Theory
Optimal branch-decomposition of planar graphs in O(n3) Time
ACM Transactions on Algorithms (TALG)
Improved Approximation Algorithms for Minimum Weight Vertex Separators
SIAM Journal on Computing
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Approximation Algorithms for Treewidth
Algorithmica
Rank-width and tree-width of H-minor-free graphs
European Journal of Combinatorics
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Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time. We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk3 log k). The previous best algorithm with a running time subexponential in k was the algorithm of Gu and Tamaki [12] with a running time of O(n1+ε log n) and an approximation ratio 1.5 + 1/ε for any ε 0. The running time of our algorithm is also better than the running time of O(f(k) · n log n) of Reed's algorithm [18] for general graphs, where f is a function exponential in k.