Optimizing Tree Decompositions in MSO

The classic algorithm of Bodlaender and Kloks [J. Algorithms, 1996] solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in mso in the following sense: for every positive integer k, there is an mso transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results [LICS 2016], this implies that for every k there exists an mso transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width. 1998 ACM Subject Classification F.4.3 Formal Languages, G.2.2 Graph Theory