  This work deals with undirected graphs that have the same betweenness
centrality for each vertex, so-called betweenness uniform graphs (or BUGs). The
class of these graphs is not trivial and its classification is still an open
problem. Recently, Gago, Coroni\v{c}ov\'a-Hurajov\'a and Madaras conjectured
that for every rational $\alpha\ge 3/4$ there exists a BUG having betweenness
centrality~$\alpha$. We disprove this conjecture, and provide an alternative
view of the structure of betweenness-uniform graphs from the point of view of
their complement. This allows us to characterise all the BUGs with betweennes
centrality at most 9/10, and show that their betweenness centrality is equal to
$\frac{\ell}{\ell+1}$ for some integer $\ell\le 9$. We conjecture that this
characterization extends to all the BUGs with betweenness centrality smaller
than~1.
