AI 中文总结
本文研究图上的$n$-凸性，明确其奇偶性相关结构性质，引入准星概念并证明归约性质，揭示其与邻域预序$P(G)$的联系，完成准阈值和阈值$n$-凸几何的分类。
AI 中文摘要
本文研究图上的邻域凸性（$n$-凸性），这是一种基于公共闭邻域闭包算子的新型有限凸空间。与标准的基于路径的图凸性不同，$n$-凸性表现出非典范行为，产生了引人注目的结构性质，且几乎不具有遗传性。聚焦于形成$n$-凸几何的图的性质，出现了奇偶性区分：一个$n$-凸几何包含一个星顶点当且仅当其顶点数为奇数。每个奇阶$n$-凸几何都可通过将一个星顶点附加到一个偶阶$n$-凸几何上唯一构造。我们引入准星的概念（度数为$\vert V\vert-2$的顶点），并证明了一个归约性质，该性质允许通过移除一对顶点（其中一个是准星）来系统地归约$n$-凸几何。最后，我们探讨了$n$-凸性与邻域预序$P(G)$之间的联系，证明$n$-凸集是$P(G)$的上集，且在无星的$n$-凸几何中，准星恰好对应$P(G)$的极大元。我们通过对准阈值和阈值$n$-凸几何进行分类完成了研究。
英文摘要
In this paper, we investigate the neighbourhood convexity ($n$-convexity) on graphs, a new finite convexity space grounded in the common closed neighbourhood closure operator. Unlike standard path-based graph convexities, $n$-convexity shows a non-canonical behaviour, giving rise to compelling structural properties and being almost never hereditary. Focusing on the properties of graphs that form $n$-convex geometries, a parity distinction emerges: an $n$-convex geometry contains a star vertex if and only if the number of its vertices is odd. Every odd-order $n$-convex geometry can be uniquely constructed by attaching a star vertex to an even-order one. We introduce the concept of quasi-stars (vertices of degree $\vert{}V\vert{}-2$) and prove a reduction property that allows systematically reducing an $n$-convex geometry by removing a pair of vertices, one of which is a quasi-star. Finally, we explore the connections between $n$-convexity and $P(G)$, the neighbourhood preorder, demonstrating that $n$-convex sets are upsets of $P(G)$ and that, in star-free $n$-convex geometries, quasi-stars correspond precisely to the maximal elements of $P(G)$. We complete our study by classifying quasi-threshold and threshold $n$-convex geometries.
