In this paper, we show that the treewidth and pathwidth of a permutati
on graph can be computed in polynomial time. In fact we show that, for
permutation graphs, the treewidth and pathwidth are equal. These resu
lts make permutation graphs one of the few nontrivial graph classes fo
r which, at the moment, treewidth is known to be computable in polynom
ial time. Our algorithm, which decides whether the treewidth (pathwidt
h) is at most some given integer Ic, can be implemented to run in O(nk
) time when the matching diagram is given. We show that this algorithm
can easily be adapted to compute the pathwidth of a permutation graph
in O(nk) time, where k is the pathwidth.