  Koh and Tay proved a fundamental classification of $G$ vertex-multiplications
into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. They
also showed that any vertex-multiplication of a tree with diameter at least 3
does not belong to the class $\mathscr{C}_2$. Of interest, $G$
vertex-multiplications are extensions of complete $n$-partite graphs and Gutin
characterised complete bipartite graphs with an ingenious use of Sperner's
Theorem. In this paper, we investigate vertex-multiplications of trees with
diameter $4$ in $\mathscr{C}_0$ (or $\mathscr{C}_1$) and exhibit its intricate
connections with problems in Sperner Theory, thereby extending Gutin's
approach. Let $s$ denote the vertex-multiplication of the central vertex. We
almost completely characterise the case of even $s$ and give a complete
characterisation for the case of odd $s\ge 3$.
