Cographs and Minimum Diamond-Generating Edge Sets in Boolean Lattices

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.