Algorithms finding tree-decompositions of graphs

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Abstract

A graph G has tree-width at most w if it admits a tree-decomposition of width ≤ w. It is known that once we have a tree-decomposition of a graph G of bounded width, many NP-hard problems can be solved for G in linear time. For w ≤ 3 we give a linear time algorithm for finding such a decomposition and for a general fixed w we obtain a probabilistic algorithm with execution time O(n log2 n + n log n∥log p∥), which for a graph G on n vertices and a real number p > 0 either finds a tree-decomposition of width ≤ 6w or answers that the tree-width of G is ≥ w; this second answer may be wrong but with probability at most p. The second result is based on a separator technique which may be of independent interest.

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This research was carried out at Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Praha 8, Czechoslovakia.

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