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Roughly, a graph has small " tree-width " if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating tree-width with the largest such obstructions (ii) an association between such obstructions and large grid(More)
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 min-cut, 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(More)