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.

翻译：顶点完整性是一种图度量，它恰好位于两个更深入研究的图概念——顶点覆盖与树深之间，且近期作为结构图参数受到关注。本文从算法元定理角度研究该参数涉及的一阶逻辑和一元二阶逻辑算法权衡。主要正面结果如下：(i) 给定顶点完整性为$k$的图$G$和含$q$个量词的一阶公式$\phi$，判定$G$是否满足$\phi$可在时间$2^{O(k^2q+q\log q)}+n^{O(1)}$内完成；(ii) 对于含$q$个量词的一元二阶公式，相同判定可在时间$2^{2^{O(k^2+kq)}}+n^{O(1)}$内完成。两项结果均通过核化论证实现，预处理将输入规模分别压缩至$2^{O(k^2)}q$和$2^{O(k^2+kq)}$。本文元定理的复杂度显著优于树深对应的元定理（涉及指数塔结构），但劣于顶点覆盖已知元定理的约$2^{O(kq)}$和$2^{2^{O(k+q)}}$复杂度。为解释此退化现象，我们构造两种公式以获得细粒度复杂度下界，并确立元定理对$k$的依赖已为最优。具体而言，我们证明在ETH假设下，无法在时间$2^{o(k^2q)}$内判定含$q$个量词的一阶公式，且存在常量规模的一元二阶公式不能在时间$2^{2^{o(k^2)}}$内判定。因此，对$k$的二次平方依赖不可避免，顶点完整性在一阶逻辑和一元二阶逻辑中的复杂度确为顶点覆盖与树深之间的中间态。