Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property. Regarding k as the parameter, a dichotomy was recently shown based on the number of quantifier alternations of first-order formulas that describe the property. In this paper, we refine this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern. The more fine-grained approach uncovers new tractable fragments, which we show to not only lie in FPT, but even in parameterized constant-depth circuit complexity classes. On the other hand, we show that vertex deletion becomes intractable already for just one quantifier per alternation, that is, there is a formula of the form {\forall}x{\exists}y{\forall}z({\psi}), with {\psi} quantifier-free, for which the vertex deletion problem is W[1]-hard. The fine-grained analysis also allows us to uncover differences in the complexity landscape when we consider different kinds of graphs and more general structures: While basic graphs (undirected graphs without self-loops), undirected graphs, and directed graphs each have a different frontier of tractability, the frontier for arbitrary logical structures coincides with that of directed graphs.

翻译：图顶点删除问题在经典与参数化复杂性理论中受到广泛研究。此类问题探讨能否从输入图中删除至多k个顶点，使得剩余图具有特定性质。将k视为参数时，近期研究基于描述该性质的一阶公式的量词交替次数建立了二分性结论。本文通过从量词交替转向独立量词模式、从二分性转向三分性，实现了基于量词模式的顶点删除问题复杂性完整分类，从而细化了该分类体系。这种更精细的研究方法揭示了新的可处理片段，我们证明这些片段不仅属于FPT类，甚至属于参数化常数深度电路复杂性类。另一方面，我们证明顶点删除问题在仅需每个交替使用一个量词时即变得难解——存在形如∀x∃y∀z(ψ)的公式（其中ψ无量词），其对应的顶点删除问题是W[1]-难的。精细分析还使我们能够发现考虑不同类型图及更一般结构时复杂性图景的差异：虽然基础图（无自环无向图）、无向图和有向图各自具有不同的可处理性边界，但任意逻辑结构的可处理性边界与有向图保持一致。