# Space-Efficient Vertex Separators for Treewidth

Practical applications that use treewidth algorithms have graphs with treewidth k = O(\sqrt[3]n). Given such n-vertex graphs we present a word-RAM algorithm to compute vertex separators using only O(n) bits of working memory. As an application of our algorithm, we show an O(1)- approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in c^k n(log^* n) log log n time using O(n) bits for some constant c. We finally show that our tree-decomposition algorithm can be used to solve several monadic second-order problems using O(n) bits as long as the treewidth of the graph is smaller than c' log n for some constant 0 < c' < 1.

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