First-order transducibility among classes of sparse graphs

TL;DR

This work analyzes first-order transductions among sparse graph classes, introducing a transduction-based hierarchy and proving three sharp non-transducibility results: between successive treewidth classes, Hadwiger-number classes, and planar versus bounded-treewidth graphs. The authors establish a key containment: if a weakly sparse class is transducible from a bounded-expansion class, then it embeds as a $k$-congested depth-$k$ minor in the source graph after adding a universal vertex, and they use the asymptotic growth of weak coloring numbers as a transduction invariant to compare classes. By relating transducibility to the growth rate of $\mathsf{wcol}_d$ (denoted $\pi_\mathscr C(d)$), they show that $\pi_\mathscr D$ cannot dominate $\pi_\mathscr C$ under transductions, enabling explicit non-transducibility results such as $\mathscr T_{t+1}\nrightarrow\mathscr T_t$, $\mathscr H_{t+2}\nrightarrow\mathscr H_t$, and $\mathscr T_4\nrightarrow\mathscr Pl$. The work highlights the asymptotic weak coloring-number behavior as a robust obstruction tool in the transduction quasi-order for sparse graphs, complementing known positive results on bounded-pathwidth and related classes.

Abstract

We prove several negative results about first-order transducibility for classes of sparse graphs: - for every $t \in \mathbb{N}$, the class of graphs of treewidth at most $t+1$ is not transducible from the class of graphs of treewidth at most $t$; - for every $t \in \mathbb{N}$, the class of graphs with Hadwiger number at most $t+2$ is not transducible from the class of graphs with Hadwiger number at most $t$; and - the class of graphs of treewidth at most $4$ is not transducible from the class of planar graphs. These results are obtained by combining the known upper and lower bounds on the weak coloring numbers of the considered graph classes with the following two new observations: - If a weakly sparse graph class $\mathscr D$ is transducible from a class $\mathscr C$ of bounded expansion, then for some $k \in \mathbb{N}$, every graph $G \in \mathscr D$ is a $k$-congested depth-$k$ minor of a graph $H^\circ$ obtained from some $H\in \mathscr C$ by adding a universal vertex. - The operations of adding a universal vertex and of taking $k$-congested depth-$k$ minors, for a fixed $k$, preserve the degree of the distance-$d$ weak coloring number of a graph class, understood as a polynomial in $d$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2: cf. sparsity and sparsityNotes
• Theorem 2.3: Zhu09
• Theorem 3.1
• Theorem 3.2: see Dreier23 and PilipczukST18a
• Lemma 3.3
• proof : Proof of \ref{['thm:trans-shallow']} using \ref{['lem:cong-on-wcols']}