arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2302.00352math.COcs.DMcs.DScs.LO
收藏

Flip-width：稠密图上的Cops and Robber

Flip-width: Cops and Robber on dense graphs

Szymon Toruńczyk

备注：
80 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文提出了新的图参数flip-width，通过Cops and Robber游戏变体统一了稀疏图与稠密图的温顺性理论，证明了其一阶逻辑下的保持性，并推测其与模型论中的monadically dependent类及固定参数可处理的模型检测问题相关。

摘要：

我们定义了新的图参数，称为flip-width，它推广了稀疏图的treewidth、degeneracy和generalized coloring numbers，以及稠密图的clique-width和twin-width。flip-width参数使用Cops and Robber游戏的变体来定义，其中robber的速度以固定常数$r\in\mathbb N\cup\{\infty\}$为界，而cops对所考虑的图执行flips（或perturbations）。然后，我们提出了一种新的图类温顺性概念，称为bounded flip-width，它是Nešetril和Ossona de Mendez的bounded expansion类的稠密对应物，并包含了Bonnet、Kim、Thomassé和Watrigant的bounded twin-width类。这统一了Sparsity Theory和Twin-width Theory，为研究这两个理论的核心概念提供了共同语言，例如weak coloring numbers和twin-width——对应于一个玩家的获胜策略——或dense shallow minors、rich divisions或well-linked sets，对应于另一个玩家的获胜策略。我们证明了flip-width的有界性在一阶解释或transductions下得到保持，推广了先前关于bounded expansion类和bounded twin-width类的结果。我们提供了一种近似给定图flip-width的算法，该算法在图大小上以切片多项式时间（XP）运行。最后，我们提出了一个更一般的温顺性概念，称为almost bounded flip-width，它是nowhere dense类的稠密对应物。我们推测并提供了证据，表明具有almost bounded flip-width的类与Shelah在模型论中引入的monadically dependent（或monadically NIP）类一致。我们还提供了证据，表明almost bounded flip-width类刻画了模型检测问题是固定参数可处理的遗传图类。

英文摘要：

We define new graph parameters, called flip-width, that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. The flip-width parameters are defined using variants of the Cops and Robber game, in which the robber has speed bounded by a fixed constant $r\in\mathbb N\cup\{\infty\}$, and the cops perform flips (or perturbations) of the considered graph. We then propose a new notion of tameness of a graph class, called bounded flip-width, which is a dense counterpart of classes of bounded expansion of Nešetril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomass{é}, and Watrigant. This unifies Sparsity Theory and Twin-width Theory, providing a common language for studying the central notions of the two theories, such as weak coloring numbers and twin-width -- corresponding to winning strategies of one player -- or dense shallow minors, rich divisions, or well-linked sets, corresponding to winning strategies of the other player. We prove that boundedness of flip-width is preserved by first-order interpretations, or transductions, generalizing previous results concerning classes of bounded expansion and bounded twin-width. We provide an algorithm approximating the flip-width of a given graph, which runs in slicewise polynomial time (XP) in the size of the graph. Finally, we propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent (or monadically NIP) classes, introduced by Shelah in model theory. We also provide evidence that classes of almost bounded flip-width characterise the hereditary graph classes for which the model-checking problem is fixed-parameter tractable.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑