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 C^k_q, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio~(2021) proved that two graphs satisfy the same C^k_q-sentences if and only if they are homomorphism indistinguishable over the class 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 the graph class T^k_q. This allows us to separate T^k_q from the intersection TW_{k-1} \cap TD_q, provided that q is sufficiently larger than k. Here TW_{k-1} is the class of all graphs of treewidth at most k-1 and TD_q is the class of all graphs of treedepth at most q. We are able to lift this separation to a (semantic) separation of the respective homomorphism indistinguishability relations \equiv_{T^k_q} and \equiv_{TW_{k-1} \cap TD_q}. We do this by showing that the classes TD_q and T^k_q are homomorphism distinguishing closed, as conjectured by Roberson~(2022). In order to prove Roberson's conjecture for T^k_q we characterise 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 show how to transform Cops' winning strategy into a pree-tree-decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first `cleaning up' procedure along the pree-tree-decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument.