Transducing paths in graph classes with unbounded shrubdepth

TL;DR

This paper characterizes when a graph class can be obtained from bounded-shrubdepth trees via FO transductions, linking it to the inability to transduce all paths, and proves a related graph decomposition theorem.

Contribution

It establishes a complete characterization of FO transducibility from bounded-shrubdepth classes in terms of path transductions, resolving an open problem in the field.

Findings

Characterization of FO transducibility in terms of path transductions.

A graph decomposition theorem for graphs excluding certain semi-induced subgraphs.

Implication that such graphs are linearly chi-bounded.

Abstract

Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class $\mathscr{C}$ can be $\mathsf{FO}$-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from $\mathscr{C}$ one cannot $\mathsf{FO}$-transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the $\mathsf{MSO}$-transduction quasi-order, even in the stronger form that concerns $\mathsf{FO}$-transductions instead of $\mathsf{MSO}$-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 $\chi$-bounded.