  We study the complexity of finding an optimal hierarchical clustering of an
unweighted similarity graph under the recently introduced Dasgupta objective
function. We introduce a proof technique, called the normalization procedure,
that takes any such clustering of a graph $G$ and iteratively improves it until
a desired target clustering of G is reached. We use this technique to show both
a negative and a positive complexity result. Firstly, we show that in general
the problem is NP-complete. Secondly, we consider min-well-behaved graphs,
which are graphs $H$ having the property that for any $k$ the graph $H(k)$
being the join of $k$ copies of $H$ has an optimal hierarchical clustering that
splits each copy of $H$ in the same optimal way. To optimally cluster such a
graph $H(k)$ we thus only need to optimally cluster the smaller graph $H$.
Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce.
We use the normalization procedure to show that also the cycle on 6 vertices is
min-well-behaved.
