Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs

In this paper we give, for all constants k, l, explicit algorithms that, given a graph Ž . G s V, E with a tree-decomposition of G with treewidth at most l, decide Ž . whether the treewidth or pathwidth of G is at most k, and, if so, find a Ž . tree-decomposition or path-decomposition of G of width at most k, and that use Ž < <. O V time. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning and are single exponential in k and l. This result can w be combined with a result of B. Reed in ‘‘Proceedings of the 24th Annual x Symposium on Theory of Computing,’’ pp. 221]228, 1992 , yielding explicit Ž . O n log n algorithms for the problem, given a graph G, to determine whether the Ž . Ž . treewidth or pathwidth of G is at most k, and, if so, to find a treeor pathŽ . w decomposition of width at most k k constant . Also, H. L. Bodlaender in ‘‘Proceedings of the 25th Annual Symposium on Theory of Computing,’’ pp. x 226]234, 1993 has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial Ž . time algorithm that, when given a graph G s V, E with treewidth F k, computes the pathwidth of G and a path-decomposition of G of minimum width. Q 1996