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 ofâ€¦ (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 allâ€¦ (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 isâ€¦ (More)

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 thatâ€¦ (More)

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give anotherâ€¦ (More)