On the treewidth and pathwidth of permutation graphs

In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[7r] is a permutation graph with treewidth k, then we show that the pathwidth of G[7r] is at most 2k, and we give an algorithm which constructs a path-decomposition with width at most 2k in time O( nk). We assume that the permutation 7r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.