CMSO obstruction characterization of linear cliquewidth

Prove that a graph class has bounded linear cliquewidth if and only if the class of trees cannot be CMSO-transduced from it.

Background

The survey presents the proposed linear analogue of the CMSO obstruction theorem for cliquewidth. Unlike the corresponding cliquewidth result, the statement is explicitly identified as unresolved.

References

An analogue of \cref{thm:cw-dependent} for linear cliquewidth would be the following statement, which to the best of our knowledge is at this point still open.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Conjecture 2.20, Section 2.3