Transducing paths in graph classes with unbounded shrubdepth

@article{Pilipczuk2022TransducingPI,
  title={Transducing paths in graph classes with unbounded shrubdepth},
  author={Michal Pilipczuk and Patrice Ossona de Mendez and Sebastian Siebertz},
  journal={ArXiv},
  year={2022},
  volume={abs/2203.16900}
}
Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class C can be 𝖥𝖮 -transduced from a class of bounded-height trees (that is, has bounded shrubdepth ) if, and only if, from C one cannot 𝖥𝖮 -transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the 𝖬𝖲𝖮 -transduction quasi-order, even in the stronger form… 

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