Let $G(n,l)$ denote the set of connected graphs with $n$ vertices and $l$ edges and let $G\_0(n,l)$ denote the elements of $G(n,l)$ without leaves. It is easily seen that $G\_0(n,n-1)$ is empty, since $G(n,n-1)$ is the set of trees, and $G\_0(n,n)$ consists of all circular grpahs of length $n$.


**Question:** How can we characterize all elements of $G\_0(n,n+1)$ (maybe also $G\_0(n,n+2)$) as belonging to some easily described classes of graphs and what would those classes look like? (in analogy to circle graphs for $G\_0(n,n)$)


Sorry for the soft question, I guess someone must have already characterized such graphs but I can't find any sources, so I would also be thankful for references to previous work in this direction.


Any help is much apprechiated.



Characterization of graphs without leaves
The elements of $G\_0(n,n+k)$ for $k\geq 1$ can be characterized as follows. Consider any partition $e\_1+e\_2+\cdots+e\_v = 2k$ of $2k$ into $v\leq n$ parts and any multigraph $S$ with $v$ vertices of degrees $2+e\_1, 2+e\_2, \ldots,2+e\_v$ and $v+k$ edges. Then any graph $H\in G\_0(n,n+k)$ is obtained by distributing $n-v$ vertices of degree two on the edges of $S$, for a suitable choice of partition and multigraph $S$. The multigraph $S$ is sometimes called the skeleton of $H$. In the case $k=1$, therefore three different multigraphs $S$ occur (pictorially: $\infty$, $\theta$ and $0\!\!-\!\!0$)


