🤖 AI Summary
This study investigates whether monotone stable graph classes that are co-matching independent can be precisely characterized via existential-positive first-order (∃⁺-FO) transductions from nowhere dense graph classes. To this end, the authors propose the Existential-Positive Sparsification Conjecture and introduce a novel combinatorial operation called subflip to capture the structural properties of such graphs. The work strengthens existing characterizations of monotone stability through flip-flatness and flip games, and establishes the expressive equivalence between ∃⁺-FO and existential-positive monadic second-order logic (MSO) over relational structures. The conjecture is verified for all known special cases, and the authors demonstrate that a sparse preimage can be directly extracted as a subgraph of the input graph, yielding an effective constructive method for sparse representations of dense graphs.
📝 Abstract
Monadic stability generalizes many tameness notions from structural graph theory such as planarity, bounded degree, bounded tree-width, and nowhere density. The sparsification conjecture predicts that the (possibly dense) monadically stable graph classes are exactly those that can be logically encoded by first-order (FO) transductions in the (always sparse) nowhere dense classes. So far this conjecture has been verified for several special cases, such as for classes of bounded shrub-depth, and for the monadically stable fragments of bounded (linear) clique-width, twin-width, and merge-width. In this work we propose the existential positive sparsification conjecture, predicting that the more restricted co-matching-free, monadically stable classes are exactly those that can be transduced from nowhere dense classes using only existential positive FO formulas. While the general conjecture remains open, we verify its truth for all known special cases of the original conjecture. Even stronger, we find the sparse preimages as subgraphs of the dense input graphs. As a key ingredient, we introduce a new combinatorial operation, called subflip, that arises as the natural co-matching-free analog of the flip operation, which is a central tool in the characterization of monadic stability. Using subflips, we characterize the co-matching-free fragment of monadic stability by appropriate strengthenings of the known flip-flatness and flipper game characterizations for monadic stability. In an attempt to generalize our results to the more expressive MSO logic, we discover (rediscover?) that on relational structures (existential) positive MSO has the same expressive power as (existential) positive FO.