Every graph class of bounded cliquewidth and unbounded linear cliquewidth contains arbitrarily large tree-like induced subgraphs that MSO-transduce all trees and FO-transduce subdivisions of all binary trees.
By flipping edges as in Claim A.71, we can assume without loss of generality that the vertex on the left side has supercolourAand the vertex on the right side has supercolourB
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Trees in graphs of large linear cliquewidth
Every graph class of bounded cliquewidth and unbounded linear cliquewidth contains arbitrarily large tree-like induced subgraphs that MSO-transduce all trees and FO-transduce subdivisions of all binary trees.