Approximating the treewidth of AT-free graphs

. Using the specic structure of the minimal separators of AT-free graphs, we give a polynomial time algorithm that computes a triangulation whose width is no more than twice the treewidth of the input graph. 1 Introduction The treewidth of graphs, introduced by Robertson and Seymour [12], has been intensively studied in the last years, mainly because many NP-hard problems become solvable in polynomial and even in linear time when restricted to graphs with small treewidth. These algorithms use a treedecomposition of small width of the graph. A tree-decomposition or a triangulation of a graph is a chordal supergraph, i.e. all the cycles of the supergraph of length strictly more than three have a chord. Computing the treewidth of a graph corresponds to nding a triangulation with the smallest cliquesize. In particular, we can restrict ourselves to triangulations minimal by inclusion, that we call minimal triangulations. Computing the treewidth of arbitrary graphs is NP-hard. Neverthele..