A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5

Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, as an extension of complete $n$-partite graphs. They proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. It was shown that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, for vertex-multiplications of trees with diameter $5$, some necessary and sufficient conditions for $\mathscr{C}_0$ were established. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $\mathscr{C}_0$ and $\mathscr{C}_1$.