Going deep and going wide: Counting logic and homomorphism indistinguishability over graphs of bounded treedepth and treewidth
This work addresses foundational questions in descriptive complexity theory for logicians and computer scientists, providing incremental advances in understanding graph homomorphism indistinguishability.
The paper tackles the problem of characterizing the expressive power of first-order logic with counting quantifiers using homomorphism indistinguishability, by separating classes of graphs based on treedepth and treewidth and proving a conjecture about homomorphism distinguishing closure, with results including a separation condition when q is sufficiently larger than k.
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.