Definable decompositions for graphs of bounded linear cliquewidth.

We prove that for every positive integer k, there exists an MSO1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some clique 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 CMSO1-definability and recognizability on graphs of bounded linear cliquewidth.