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

Application of some techniques in Sperner Theory: Optimal orientations of vertex-multiplications of trees with diameter 4