Finding approximate separators and computing tree width quickiy

Abstract

We show that for any fixed κ, there is a linear-time algorithm which given a graph G either: (i) finds a cutset X of G with |X|≤ κ such that no component of G-X contains more than 3/4|G-X| vertices, or (ii) determines that for any set X of vertices of G with |X| ≤ κ, there is a component of G - X which contains more than 2/3|G - X\ vertices. This approximate separator algorithm can be used to develop an 0(n log n) algorithm for determining if G has a tree decomposition of width at most k (for fixed k) and finding such a tree decomposition if it exists.