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Graph Minors .XIII. The Disjoint Paths Problem
An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
Graph Minors. II. Algorithmic Aspects of Tree-Width
Graph minors. V. Excluding a planar graph
Quickly Excluding a Planar Graph
A much better bound is proved on the tree-width of planar graphs with no minor isomorphic to a g × g grid and this is the best known bound.
The Strong Perfect Graph Theorem
A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of…
Graph minors. X. Obstructions to tree-decomposition
- Thor Johnson, N. Robertson, P. Seymour, Robin Thomas
- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 1 May 2001
It is proved that every directed graph with no “haven” of large order has small tree-width, and the Hamilton cycle problem and other NP-hard problems can be solved in polynomial time when restricted to digraphs of bounded tree- width.
The Four-Colour Theorem
- N. Robertson, Daniel P. Sanders, P. Seymour, Robin Thomas
- MathematicsJ. Comb. Theory, Ser. B
- 1 May 1997
Another proof is given, still using a computer, but simpler than Appel and Haken's in several respects, that every loopless planar graph admits a vertex-colouring with at most four different colours.
Graph Minors. XVI. Excluding a non-planar graph