A linear time algorithm tree-decompositions for finding of small treewidth* Hans L. Bodlaendert Abstract In this paper, we give, for constant k, a linear time algorithm, that given a graph G = (V, E), the treewidth determines whether of G is at most k, and if so, finds a tree- far, this step dominated the running time of most algo- rithms, as the second step (some kind of dynamic programming algorithm using the tree-decomposition) usually costs only linear time. The best algorithm known so far for thk first step was an algorithm by Reed , which costs O(n log n). In this paper, we improve on this result, and give a linear time algorithm. The problem Given a graph G = (V, E) and an integer k, decomposition of G with treewidth at most k. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm. Keywords: graph algorhhms, treewidth, pathwidth, partial Mrees, graph minors. is the treewidth of G at most k is NP-complete . Much work has been done on this problem for constant k. For k = 1,2,3, linear time algorithms
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