布尔格中的余图与最小生成钻石边集
Cographs and Minimum Diamond-Generating Edge Sets in Boolean Lattices
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中文总结 AI 辅助
研究布尔格覆盖边上局部闭包运算，证明生成全覆盖图的覆盖边集基数至少\(n\)并分类。指出图\(G\)相关集合\(S_G\)钻石生成全覆盖图当且仅当\(G\)是余图，最小基数生成元唯一，其带标签集与\(n\)顶点带标签余图一一对应。
中文摘要 AI 辅助
我们研究布尔格覆盖边上的一种局部闭包运算：当正方形面的两条下边或两条上边存在时，该正方形的四条边都被添加。我们证明，生成\(\mathcal{B}_n\)全覆盖图的每个覆盖边集的基数至少为\(n\)，并对达到此界的所有生成元进行分类。对于\([n]\)上的图\(G\)，令\(S_G = \{N_G(i) \to N_G(i) \cup \{i\} : i \in [n]\}\)。那么当且仅当\(G\)是余图时，\(S_G\)钻石生成全覆盖图，且每个最小基数生成元都以这种方式唯一产生。因此，\(\mathcal{B}_n\)的带标签最小生成钻石集与\(n\)个顶点的带标签余图一一对应。
英文摘要
We study a local closure operation on the cover edges of a Boolean lattice: whenever the two lower edges or the two upper edges of a square face are present, all four edges of that square are added. We prove that every set of cover edges generating the full cover graph of $\mathcal{B}_n$ has cardinality at least $n$, and we classify all generators attaining this bound. For a graph $G$ on $[n]$, let $S_G=\{N_G(i)\to N_G(i)\cup\{i\}:i\in[n]\}$. Then $S_G$ diamond-generates the full cover graph if and only if $G$ is a cograph, and every minimum-cardinality generator arises uniquely in this way. Consequently, labeled minimum diamond-generating sets of $\mathcal{B}_n$ are in bijection with labeled cographs on $n$ vertices.
