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.

翻译：一元稳定性概括了结构图论中的许多驯服性概念，如平面性、有界度、有界树宽和无处稠密性。稀疏化猜想预测，（可能稠密的）一元稳定图类正是那些可以通过一阶逻辑转换在（总是稀疏的）无处稠密类中进行逻辑编码的图类。迄今为止，该猜想已在多个特殊情况下得到验证，例如有界灌木深度的图类，以及有界（线性）团宽、双宽和合并宽的一元稳定片段。在本工作中，我们提出了存在正性稀疏化猜想，预测更受限的无共匹配一元稳定类正是那些仅使用存在正性一阶公式从无处稠密类转换而来的图类。尽管一般猜想仍然开放，但我们验证了其在原始猜想所有已知特殊情况下的真实性。甚至更强地，我们找到了作为稠密输入图子图的稀疏原像。作为一个关键要素，我们引入了一种新的组合操作，称为子翻转，它作为翻转操作的自然无共匹配类比而出现，而翻转操作是表征一元稳定性的核心工具。利用子翻转，我们通过适当强化已知的翻转平坦性和翻转游戏表征来刻画一元稳定性的无共匹配片段。在尝试将我们的结果推广到更具表达力的MSO逻辑时，我们发现（重新发现？）在关系结构上，（存在）正性MSO与（存在）正性FO具有相同的表达能力。