Almost-2-regular random graphs

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.