Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth

TL;DR

This paper analyzes the expressive power of the counting logic fragment $\mathsf{C}^{k}_{q}$ via homomorphism indistinguishability, establishing that $\mathsf{C}^{k}_{q}$-equivalence corresponds to indistinguishability over the graph class $\mathcal{T}^{k}_{q}$ of graphs with a $k$-pebble forest cover of depth $q$. It then shows that $\mathcal{T}^{k}_{q}$ is strictly contained in the intersection of treewidth and treedepth classes $\mathcal{TW}_{k-1} \cap \mathcal{TD}_{q}$ for sufficiently large $q$, and that both $\mathcal{TD}_{q}$ and $\mathcal{T}^{k}_{q}$ are homomorphism-distinguishing closed. A central contribution is the Cops-and-Robber game characterization of $\mathcal{T}^{k}_{q}$ through a monotone strategy, implemented via pre-tree-decompositions and a width-decreasing cleaning procedure. The results yield semantic separations between the associated indistinguishability relations and extend the landscape of guarded fragments by relating GC$^{k}_{q}$ to a related graph class $\mathcal{G}\mathcal{T}^{k}_{q}$. Overall, the work clarifies the limits of using minor- and depth-based decompositions to capture counting-logic expressivity and points to connections with Weisfeiler–Leman-type algorithms.

Abstract

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $\mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $\mathsf{C}^k_q$-sentences iff they are homomorphism indistinguishable over the class $\mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. After reproving this result using elementary means, we provide a graph-theoretic analysis of $\mathcal{T}^k_q$. This allows us to separate $\mathcal{T}^k_q$ from the intersection $\mathcal{TW}_{k-1} \cap \mathcal{TD}_q$, provided that $q$ is sufficiently larger than $k$. Here $\mathcal{TW}_{k-1}$ is the class of all graphs of treewidth at most $k-1$ and $\mathcal{TD}_q$ is the class of all graphs of treedepth at most $q$. We are able to lift this separation to a separation of the respective homomorphism indistinguishability relations $\equiv_{\mathcal{T}^k_q}$ and $\equiv_{\mathcal{TW}_{k-1} \cap \mathcal{TD}_q}$. We do this by showing that the classes $\mathcal{TD}_q$ and $\mathcal{T}^k_q$ are homomorphism distinguishing closed, as conjectured by Roberson (2022). In order to prove Roberson's conjecture for $\mathcal{T}^k_q$, we characterise $\mathcal{T}^k_q$ in terms of a monotone Cops-and-Robber game. The crux is to prove that if Cop has a winning strategy then Cop also has a winning strategy that is monotone. To that end, we transform Cops' winning strategy into a pree-tree-decomposition, which is inspired by decompositions of matroids, and then apply an intricate breadth-first cleaning up procedure along the pree-tree-decomposition (which may temporarily lose the property of representing a strategy). Thereby, we achieve monotonicity while controlling the number of rounds across all branches of the decomposition via a vertex exchange argument.

Figures (6)

• Figure 1: Tree-decomposition and forest cover of grids. (a) A tree-decomposition of the grid $G_{2 \times 5}$ of width $3$. (b) A forest cover of the grid $G_{2 \times 7}$ of depth $6$. The edges of the original grid are dashed.
• Figure 2: A $4$-construction tree for the grid $G_{2 \times 7}$ of elimination depth $6$. Edges entering elimination nodes are dashed.
• Figure 3: Graph and corresponding pre-tree-decomposition from \ref{['ex:pre-tree-dec']}.
• Figure 4: The subtree $T_i$ appearing in the construction.
• Figure 5: Example of the construction, see \ref{['ex:monotony-construction']}.

Theorems & Definitions (107)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.4
• Theorem 1.5
• definition 1
• Lemma 2.1
• definition 2
• Lemma 2.2