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

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.