Transducing paths in graph classes with unbounded shrubdepth
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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 that concerns 𝖥𝖮-transductions instead of 𝖬𝖲𝖮-transductions. The backbone of our proof is a graph-theoretic statement that says the following: If a graph G excludes a path, the bipartite complement of a path, and a half-graph as semi-induced subgraphs, then the vertex set of G can be partitioned into a bounded number of parts so that every part induces a cograph of bounded height, and every pair of parts semi-induce a bi-cograph of bounded height. This statement may be of independent interest; for instance, it implies that the graphs in question form a class that is linearly χ-bounded.
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