We prove that for every positive integer k, there exists an MSO_1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of k. A direct corollary of this result is the equivalence of the notions of CMSO_1-definability and recognizability on graphs of bounded linear cliquewidth.

@article{paperbot1346,
  title = {Definable decompositions for graphs of bounded linear cliquewidth},
  author = {Mikołaj Bojańczyk and Martin Grohe and Michał Pilipczuk},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 1},
  year = {2021},
  doi = {10.23638/lmcs-17(1:5)2021}
}