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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 + 1 vertices. We prove that this equals the maximum k such that there is a collection of connected subgraphs, pairwise intersecting or adjacent, such that no set of ≤ k vertices meets all of them. A corollary is an analogue of… (More)

A graph is said to be k-linked if it has at least 2k vertices and for every sequence of distinct vertices there exist disjoint paths P1,. .. , P k such that the ends of Pi are si and ti. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive… (More)

We generalize the concept of tree-width to directed graphs, and prove that every directed graph with no " haven " of large order has small tree-width. Conversely, a digraph with a large haven has large tree-width. We also show that the Hamilton cycle problem and other NP-hard problems can be solved in polynomial time when restricted to digraphs of bounded… (More)

Staton proved that every triangle-free graph on n vertices with maximum degree three has an independent set of size at least 5n=14. A simpler proof was found by Jones. We give a yet simpler proof, and use it to design a linear-time algorithm to nd such an independent set. Let G be a triangle-free graph on n vertices with maximum degree three. By Brooks'… (More)

We prove that, for every positive integer k, there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k-spoke wheel or K 3,k ; and that every internally 4-connected graph with at least N vertices has a minor isomorphic to the 2k-spoke double wheel, the k-rung circular ladder, the k-rung Möbius ladder, or… (More)