Going Deep and Going Wide: Counting Logic and Homomorphism Indistinguishability over Graphs of Bounded Treedepth and Treewidth
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We study the expressive power of first-order logic with counting quantifiers, especially the k-variable and quantifier-rank-q fragment ๐ข^k_q, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same ๐ข^k_q-sentences if and only if they are homomorphism indistinguishable over the class ๐ฏ^k_q 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 Dvoลรกk (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 ๐ฏ^k_q. This allows us to separate ๐ฏ^k_q from the intersection of the graph class ๐ฏ๐ฒ_k-1, that is graphs of treewidth less or equal k-1, and ๐ฏ๐_q, 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 ๐ฏ๐_q is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).
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