Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph $G$ of vertex integrity $k$ and an FO formula $\phi$ with $q$ quantifiers, deciding if $G$ satisfies $\phi$ can be done in time $2^{O(k^2q+q\log q)}+n^{O(1)}$; (ii) for MSO formulas with $q$ quantifiers, the same can be done in time $2^{2^{O(k^2+kq)}}+n^{O(1)}$. Both results are obtained using kernelization arguments, which pre-process the input to sizes $2^{O(k^2)}q$ and $2^{O(k^2+kq)}$ respectively. The complexities of our meta-theorems are significantly better than the corresponding meta-theorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly $2^{O(kq)}$ and $2^{2^{O(k+q)}}$ complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on $k$ is best possible. More precisely, we show that it is not possible to decide FO formulas with $q$ quantifiers in time $2^{o(k^2q)}$, and that there exists a constant-size MSO formula which cannot be decided in time $2^{2^{o(k^2)}}$, both under the ETH. Hence, the quadratic blow-up in the dependence on $k$ is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.

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顶点完整性的细粒度元定理

顶点完整性是一种图度量，它正好位于两个更深入研究的概念之间，即顶点覆盖和树深度，并且最近作为结构图参数受到了关注。在本文中，我们从一阶 (FO) 和一元二阶 (MSO) 逻辑的算法元定理的角度研究了与此参数相关的算法权衡。我们的积极结果如下：（i）给定顶点完整性 $k$ 的图 $G$ 和带有 $q$ 量词的 FO 公式 $\phi$，确定 $G$ 是否满足 $\phi$ 可以在时间 $2^{O(k^2q+q\log q)}+n^{O(1)}$; (ii) 对于带有 $q$ 量词的 MSO 公式，同样可以在 $2^{2^{O(k^2+kq)}}+n^{O(1)}$ 时间内完成。这两个结果都是使用内核化参数获得的，分别将输入预处理为 $2^{O(k^2)}q$ 和 $2^{O(k^2+kq)}$ 大小。我们的元定理的复杂性明显优于树深度的相应元定理，后者涉及指数塔。然而，它们比以顶点覆盖的相应元定理已知的大约 $2^{O(kq)}$ 和 $2^{2^{O(k+q)}}$ 复杂性差。为了解释这种恶化，我们提出了两个公式结构，它们导致细粒度复杂性下界，并确定我们的元定理对 $k$ 的依赖性是最好的。更准确地说，我们表明不可能在时间 $2^{o(k^2q)}$ 中确定具有 $q$ 量词的 FO 公式，并且存在无法在时间 $2 中确定的恒定大小的 MSO 公式^{2^{o(k^2)}}$，均在 ETH 下。因此，