Transducing paths in graph classes with unbounded shrubdepth

Transductions are a general formalism for expressing transformations of graphs (and more generally, relational structures) in logic. We prove that graph class C can be FO-transduced from bounded-height trees (that is, has bounded shrubdepth) if, and only one cannot FO-transduce the all paths. This establishes three remaining open questions posed by Blumensath Courcelle about MSO-transduction quasi-order, even stronger form concerns FO-transductions instead MSO-transductions. The backbone our proof is graph-theoretic statement says following: If G excludes path, bipartite complement half-graph as semi-induced subgraphs, then vertex set partitioned into number parts so every part induces cograph height, pair semi-induce bi-cograph height. may independent interest; instance, it implies question linearly χ-bounded.