AI 中文总结
本研究将余图族的良拟序范畴化结论扩展至余图顶点集多项式环，证明相关理想的普适性结果，并对余图相关拓扑与超平面排列给出约束。
AI 中文摘要
若有限简单图G不含4顶点路径P₄作为诱导子图，则称其为余图（cograph）。经典结论表明，余图族在诱导子图关系下是良拟序的[D]。在Knudsen与第三作者的前期工作[KR，定理7.2]中，已证明该良拟序命题存在范畴化，这使得作者们能证明关于余图上构形空间同调群的普适有限生成命题[KR，定理1.5]。本研究将[KR，定理7.2]扩展为与余图顶点集上的多项式环兼容的形式，由此可证明一系列与这些多项式环的边理想、环面理想相关的普适性结果，部分推广并拓展了Kahle[kahle2019binomial]的工作。此外，本研究还对余图相关的图复形、锚定构形空间可能产生的拓扑类型施加了严格限制，同时对余图超平面 arrangement 可能的组合结构给出了组合约束。
英文摘要
A finite simple graph $G$ is called a cograph if it does not contain the path on four vertices $P_4$ as an induced subgraph. It is classically known that the family of cographs are well-quasi-ordered by the induced subgraph relation \cite{D}. In preceding work of Knudsen and the third author \cite[Theorem 7.2]{KR}, it was shown that this well-quasi-order statement admitted a categorification, which allowed those authors to prove universal finite generation statements about the homology groups of configuration spaces on cographs \cite[Theorem 1.5]{KR}. In this work, we expand \cite[Theorem 7.2]{KR} to be compatible with the family of polynomial rings on the vertex sets of cographs. By consequence, we are able to prove a number of universality results related with edge and toric ideals of these polynomial ring, partially generalizing and expanding upon work of Kahle \cite{kahle2019binomial}. We also conclude strong restrictions on the kinds of topologies that can arise from graph complexes and anchored configuration spaces associated to cographs, as well as combinatorial constraints on the possible combinatorics of hyperplane arrangements of cographs.
