  A $\beta$-skeleton, $\beta \geq 1$, is a planar proximity undirected graph of
an Euclidean points set, where nodes are connected by an edge if their
lune-based neighbourhood contains no other points of the given set. Parameter
$\beta$ determines the size and shape of the lune-based neighbourhood. A
$\beta$-skeleton of a random planar set is usually a disconnected graph for
$\beta>2$. With the increase of $\beta$, the number of edges in the
$\beta$-skeleton of a random graph decreases. We show how to grow stable
$\beta$-skeletons, which are connected for any given value of $\beta$ and
characterise morphological transformations of the skeletons governed by $\beta$
and a degree of approximation. We speculate how the results obtained can be
applied in biology and chemistry.
