The dominating-set reformulation of the half-minimum-degree conjecture

Let $G$ be a graph, with vertex set $V(G)$, minimum degree $\delta(G)$, open neighborhoods $N(u)$, and a set called dominating when it meets or is adjacent to every vertex of $G$.

Dominating-set reformulation. If

$$\delta(G)\geq \frac{1}{2}|V(G)|,$$

then there are vertices $u$ and $v$ in $G$ such that

$$\{u,v\}\cup\left(N(u)\cap N(v)\right)$$

is a dominating set.

The source presents this as a reformulation of the preceding restricted-pebbling conjecture, using the fact that graphs with minimum degree at least half their order have diameter at most two and invoking earlier results. It is therefore not an independent mathematical claim, but a domination-theoretic equivalent formulation.