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.

翻译：图的翻转是通过补全顶点集合内部的边关系得到的。翻转通常用于通过增大顶点间的距离来分离图中的顶点。我们证明，在$K_{t,t}$-free图中，每个短翻转序列都可以用一个能实现类似分离程度的短顶点删除序列来模拟：所得图中的距离，在最多三倍的因子内，至少与翻转后获得的距离一样大。这一结果为结构图论和有限模型论中一个新兴模式提供了简单而统一的解释：稠密图驯服性概念的$K_{t,t}$-free片段通常与稀疏图的驯服性概念相一致。作为直接应用，我们恢复了以下已知等价关系。在$K_{t,t}$-free设定下，稠密概念（1）有界灌木深度，（2）有界团宽度，（3）有界翻转宽度，（4）单子依赖性，分别对应于稀疏概念（1）有界树深度，（2）有界树宽度，（3）有界扩张，以及（4）无处稠密性。此外，我们重新证明了Dreier和Toruńczyk（STOC 2025）的结果，即有界合并宽度的$K_{t,t}$-free类具有有界扩张性。我们的证明提供了显式界并且是直接的，因为它展示了如何从合并序列（有界合并宽度的见证）构造强染色序（有界扩张性的见证）。在此过程中，我们识别了一个新的图参数族，称为分离宽度，它夹在强染色数和弱染色数之间，并且与合并宽度参数密切相关。我们提供的证据表明，这个在图论文献中显然被忽视的图参数族，可能在稀疏图的研究中发挥基础性作用。