For an unweighted graph on $k$ terminals, Kratsch and Wahlstr\"om constructed a vertex sparsifier with $O(k^3)$ vertices via the theory of representative families on matroids. Since their breakthrough result in 2012, no improvement upon the $O(k^3)$ bound has been found. In this paper, we interpret Kratsch and Wahlstr\"om's result through the lens of Bollob\'as's Two-Families Theorem from extremal combinatorics. This new perspective allows us to close the gap for directed acyclic graphs and obtain a tight bound of $\Theta(k^2)$. Central to our approach is the concept of skew-symmetry from extremal combinatorics, and we derive a similar theory for the representation of skew-symmetric families that may have future applications.

中文翻译：

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有向无环图中顶点切割稀疏器的紧界

对于在$ k $终端上的未加权图，Kratsch和Wahlstr \“ om通过拟阵的代表族理论构造了具有$ O（k ^ 3）$顶点的顶点稀疏器。自2012年取得突破以来，对已经找到了$ O（k ^ 3）$界。在本文中，我们通过极值组合论的Bollob's的二族定理来解释Kratsch和Wahlstr'om的结果。这种新的视角使我们能够缩小有向无环图的间隙，并获得$ \ Theta（k ^ 2）$的紧密边界。我们的方法的核心是来自极值组合器的偏斜对称性概念，并且我们推导了一个类似的理论来表示偏斜对称族，这些族可能会有未来的应用。