2022/03/31 by Pilipczuk, Michał, de Mendez, Patrice Ossona, Siebertz, Sebastian · 2 citations
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.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 \mathscrC can be FO-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from \mathscrC 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.