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 mathsfCkq, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same mathsfCkq-sentences if and only if they are homomorphism indistinguishable over the class Tkq of graphs admitting a k-pebble forest cover of depth q. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvovr'ak (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class Tkq. This allows us to separate Tkq from the intersection of the graph class TWk-1, that is graphs of treewidth less or equal k-1, and TDq, that is graphs of treedepth at most q if q is sufficiently larger than k. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class TDq is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).