Theorem: Eight unlabelled connected skeletons

Every nontrivial connected loopless contraction multigraph with at most three edges has one of the following eight skeletons:

1. one edge;

2. two parallel edges;

3. a two-edge path;

4. three parallel edges;

5. a two-edge path with one edge doubled;

6. a triangle;

7. a three-edge path;

8. a three-edge star.

Every graph with at most three edges is a disjoint union of these, with total edge count at most three, together with isolated vertices.

Proof. With one edge there is one connected graph. With two edges there are either two vertices, giving parallel edges, or three vertices, giving a path.

A connected three-edge graph has at most four vertices. On two vertices it has three parallel edges. On three vertices it either joins all three pairs, giving a triangle, or joins two pairs with multiplicities two and one. On four vertices it is a tree, hence either a path or a star.

The final statement follows by decomposition into connected components.