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 the 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.

翻译：顶点完整性是一种图度量，它恰好介于两个研究更深入的概念——顶点覆盖和树深度——之间，并且最近作为一种结构图参数受到关注。本文从一阶逻辑（FO）和单子二阶逻辑（MSO）的算法元定理角度，研究了涉及该参数的算法权衡。我们的积极结果如下：（i）给定一个顶点完整性为 $k$ 的图 $G$ 和一个具有 $q$ 个量词的一阶逻辑公式 $\phi$，判定 $G$ 是否满足 $\phi$ 可以在 $2^{O(k^2q+q\log q)}+n^{O(1)}$ 时间内完成；（ii）对于具有 $q$ 个量词的 MSO 公式，同样可以在 $2^{2^{O(k^2+kq)}}+n^{O(1)}$ 时间内完成。这两个结果都是通过核化论证获得的，该论证将输入预处理到大小分别为 $2^{O(k^2)}q$ 和 $2^{O(k^2+kq)}$。我们的元定理的复杂度显著优于树深度对应的元定理（后者涉及指数塔）。然而，它们比顶点覆盖对应的元定理已知的大约 $2^{O(kq)}$ 和 $2^{2^{O(k+q)}}$ 复杂度要差。为了解释这种性能下降，我们提出了两个公式构造，它们导致了细粒度复杂度下界，并证明我们的元定理对 $k$ 的依赖是最优的。更准确地说，我们证明在 ETH 假设下，不可能在 $2^{o(k^2q)}$ 时间内判定具有 $q$ 个量词的一阶逻辑公式，并且存在一个恒定大小的 MSO 公式，不能在 $2^{2^{o(k^2)}}$ 时间内判定。因此，对 $k$ 依赖的二次增长是不可避免的，并且顶点完整性对于 FO 和 MSO 逻辑的复杂度确实介于顶点覆盖和树深度之间。