Paul D. Seymour

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In the Multiterminal Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. We show that the problem(More)
We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of(More)
The tree-width of a graph G is the minimum k such that G may be decomposed into a “treestructure” 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)
In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t+1 vertices is t-colourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is(More)
Let us regard a graph as a system of tunnels containing a (lucky, invisible, fast) fugitive. We desire to capture this fugitive by “searching” all edges of the graph, in a sequence of discrete steps, while using the fewest possible “guards.” This problem was introduced by Breisch [2] and Parsons [6]. In the version of graph searching considered in [51(More)