稀疏图中的翻转与合并宽度
Flips and Merge-Width in Sparse Graphs
AI总结：
该研究通过翻转和合并宽度分析稀疏图的结构，揭示了$K_{t,t}$-free图中密集与稀疏图性质的等价性，并提出新的图参数分离宽度。
摘要：
通过在一组顶点内翻转边关系来获得图的翻转。翻转通常用于将顶点分离，通过增加它们之间的距离。我们证明在$K_{t,t}$-free图中，任何短的翻转序列都可以通过短的顶点删除序列来模拟，以达到相似的分离程度：在得到的图中，距离至少是翻转后距离的三倍。这一结果为结构图论和有限模型理论中出现的一种新兴模式提供了简单而统一的解释：$K_{t,t}$-free片段的密集图的 tame 性质往往与稀疏图的 tame 性质一致。作为直接应用，我们恢复了已知的等价关系。在$K_{t,t}$-free设定中，密集的性质（1）有界 shrub-depth，（2）有界 clique-width，（3）有界 flip-width，（4）单调依赖，分别与稀疏的性质（1）有界树深度，（2）有界树宽，（3）有界扩展，和（4）无处密集性一致。此外，我们重新证明了 Dreier 和 Toruńczyk（STOC 2025）的结果，即$K_{t,t}$-free类的有界合并宽度具有有界扩展。我们的证明提供了明确的界限，并且是直接的，因为它展示了如何通过合并序列（有界合并宽度的见证）来构造强着色顺序（有界扩展的见证）。在过程中，我们识别出一个新的图参数家族，称为分离宽度，它介于强和弱着色数之间，并且与合并宽度参数密切相关。我们提供证据表明，这个似乎在文献中被忽视的图参数家族可能在稀疏图研究中起根本作用。
英文摘要：
A flip of a graph is obtained by complementing the edge relation within a set of vertices. Flips are typically used to separate vertices in a graph, by increasing the distances between them. We show that in $K_{t,t}$-free graphs, every short sequence of flips can be simulated by a short sequence of vertex deletions that achieves a similar degree of separation: distances in the resulting graph are, up to a factor of three, at least as large as those obtained after the flips. This result provides a simple and uniform explanation of an emerging pattern in structural graph theory and finite model theory: the $K_{t,t}$-free fragment of a tameness notion for dense graphs often coincides with a tameness notion for sparse graphs. As immediate applications, we recover the following known equivalences. In the $K_{t,t}$-free setting, the dense notions (1) bounded shrub-depth, (2) bounded clique-width, (3) bounded flip-width, (4) monadic dependence, respectively, coincide with the sparse notions (1) bounded tree-depth, (2) bounded tree-width, (3) bounded expansion, and (4) no-where dense-ness. Furthermore, we reprove the result by Dreier and Toruńczyk (STOC 2025) stating that $K_{t,t}$-free classes of bounded merge-width have bounded expansion. Our proof provides explicit bounds and is direct, as it shows how to construct strong coloring orders (witnesses of bounded expansion) from merge sequences (witnesses of bounded merge-width). Along the way, we identify a new family of graph parameters, dubbed separation-width, that is sandwiched between the strong and weak coloring numbers, and is closely related to the merge-width parameters. We provide evidence that this family of graph parameters, apparently overlooked in the literature, may play a fundamental role in the study of sparse graphs.
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