  We study a special case of the configuration model, in which almost all the
vertices of the graph have degree $2$. We show that the graph has a very
peculiar and interesting behaviour, in particular when the graph is made up by
a vast majority of vertices of degree $2$ and a vanishing proportion of
vertices of higher degree, the giant component contains $n(1-o(1))$ vertices,
but the second component can still grow polynomially in $n$. On the other hand,
when almost all the vertices have degree $2$ except for $o(n)$ which have
degree $1$, there is no component of linear size.

Probability;Combinatorics