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. We also show that mso transductions can be implemented in linear fixed-parameter time, which enables us to derive the algorithmic result of Bodlaender and Kloks as a corollary of our main result.

@article{paperbot1727,
  title = {Optimizing tree decompositions in MSO},
  author = {Mikołaj Bojańczyk and Michał Pilipczuk},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 18, Issue 1},
  year = {2022},
  doi = {10.46298/lmcs-18(1:26)2022}
}