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Call routing and the ratcatcher
TLDR
It follows that branch-width is polynomially computable for planar graphs—that too is NP-hard for general graphs.
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Quickly Excluding a Planar Graph
TLDR
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.
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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…
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Graph Searching and a Min-Max Theorem for Tree-Width
The tree-width of a graph G is the minimum k such that G may be decomposed into a "tree-structure" of pieces each with at most k + l vertices. We prove that this equals the maximum k such that there…
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Directed Tree-Width
TLDR
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.
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The Four-Colour Theorem
TLDR
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.
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A separator theorem for nonplanar graphs
Let G be an n-vertex graph with no minor isomorphic to an h- vertex complete graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A…
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Hadwiger's conjecture forK6-free graphs
TLDR
It is shown (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex.
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A separator theorem for graphs with an excluded minor and its applications
TLDR
It follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and solve any sparse system of n linear equations in n unknowns in time O(n).
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On the complexity of finding iso- and other morphisms for partial k-trees
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