发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出翻转可打包性概念，证明其分别等价于图类的一元稳定性和一元依赖性（$m=2$时），并统一了八个稀疏/稠密图类的刻画，其中参数$m$区分深度与宽度类概念。
AI 中文摘要
一个图类如果不能用固定的第一阶公式在该类的着色图中编码所有图，则称为一元依赖的；如果甚至不能编码任意长的线性序，则称为一元稳定的。Bonnet等人（ICALP 2025）通过翻转可分性刻画了一元依赖性：对于每个顶点权重，有界多次翻转——在顶点子集内对邻接关系进行补操作——使得每个半径为$r$的球至多携带$\varepsilon$-比例的权重，从而每个携带$\varepsilon$-比例的集合都有两个元素被拉开。我们引入翻转可打包性：有界多个图，每个图通过对输入进行有界多次翻转得到，且所有图在呈现任何集合之前由权重决定，使得每个携带$\varepsilon$-比例的权重的集合在其中某个图中具有$m$个两两相距很远的元素。产生每个图的翻转次数仅取决于半径；只有图的数量取决于$\varepsilon$和$m$。我们证明一个图类是翻转可打包的当且仅当它是一元稳定的，并且$2$-翻转可打包的（即$m=2$时的翻转可打包性）当且仅当它是一元依赖的。从两个分散元素到$m$个的过渡正是区分这两个概念的关键。对于一元稳定的图类，我们证明翻转后的图可以在立方时间内从权重计算出来。改变定义的三个参数——稀疏化操作、半径和分散元素的数量$m$——从同一模板产生八个已知的稀疏和稠密图类的刻画。在每种情况下，$m$区分了深度类概念与其宽度类松弛：树深度与树宽，灌木深度与团宽，以及一元稳定性与一元依赖性。
英文摘要
A class of graphs is monadically dependent if one cannot encode all graphs in coloured graphs from the class using a fixed first-order formula, and monadically stable if one cannot even encode arbitrarily long linear orders. Bonnet et al. (ICALP 2025) characterised monadic dependence by flip-separability: for every vertex weighting, boundedly many flips - complementations of the adjacency relation within a vertex subset - make every ball of radius $r$ carry at most an $\varepsilon$-fraction of the weight, so that every set carrying an $\varepsilon$-fraction has two elements pulled apart. We introduce flip-packability: boundedly many graphs, each obtained from the input by boundedly many flips and all determined by the weighting before any set is presented, such that every set carrying an $\varepsilon$-fraction of the weight has $m$ elements pairwise far apart in one of them. The number of flips producing each graph depends on the radius alone; only the number of graphs depends on $\varepsilon$ and $m$. We prove that a class of graphs is flip-packable if and only if it is monadically stable, and $2$-flip-packable, that is, flip-packable with $m=2$, if and only if it is monadically dependent. The passage from two scattered elements to $m$ is thus exactly what separates the two notions. For monadically stable classes we show that the flipped graphs can be computed from the weighting in cubic time. Varying the three parameters of the definition - the sparsifying operation, the radius, and the number $m$ of elements scattered - produces eight known characterisations of sparse and dense graph classes from the same template. In each case $m$ separates a depth-like notion from its width-like relaxation: treedepth from treewidth, shrubdepth from cliquewidth, and monadic stability from monadic dependence.
Comments32 pages; added funding information
