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，它恰好介于两个更深入研究的图概念（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。本文从一阶逻辑和一元二阶逻辑算法元定理的角度，研究了该参数涉及的算法权衡。我们的正面结果如下：（i）给定一个顶点完整性为$k$的图$G$和一个包含$q$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）对于包含$q$个量词的一元二阶公式，可在时间$2^{2^{O(k^2+kq)}}+n^{O(1)}$内完成相同任务。这两个结果均通过核化论证获得，该论证将输入分别预处理至规模$2^{O(k^2)}q$和$2^{O(k^2+kq)}$。我们的元定理的复杂度显著优于涉及指数塔的树深度对应元定理，但次于顶点覆盖已知的约$2^{O(kq)}$和$2^{2^{O(k+q)}}$复杂度。为解释此退化现象，我们构造了两种公式，导致细粒度复杂度下界，并确立了元定理对$k$的依赖是最优的。更精确地说，我们证明：在ETH假设下，不可能在时间$2^{o(k^2q)}$内判定包含$q$个量词的一阶公式，且存在一个常数大小的一元二阶公式无法在时间$2^{2^{o(k^2)}}$内判定。因此，对$k$依赖的二次增长不可避免，且顶点完整性在一阶和一元二阶逻辑中的复杂度真正介于顶点覆盖与树深度之间。