Transducing paths in graph classes with unbounded shrubdepth
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 boundedheight 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 quasiorder, even in the stronger form that concerns $\mathsf{FO}$transductions instead of $\mathsf{MSO}$transductions. The backbone of our proof is a graphtheoretic statement that says the following: If a graph $G$ excludes a path, the bipartite complement of a path, and a halfgraph as semiinduced 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 semiinduce a bicograph 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 arXiv:
 arXiv:2203.16900
 Bibcode:
 2022arXiv220316900P
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Computer Science  Logic in Computer Science;
 Mathematics  Logic