Is there an established name for graphs, that can be decomposed into 


* a tree with at least three leaf nodes and
* a connected two-regular graph with the tree's leaf nodes as vertices?


examples of those graphs are the edge-graphs of polyhedra with one facet, that is edge-adjacent to all other facets.



Name for Biconnected Tree+Cycle Graph
Ok, since it is close enough for the OP (as evidenced by the comments) I will transfer my comment to an answer so that the question can be neatly wrapped up.


So a **Halin graph** (named after Rudolf Halin) is built from a tree with no vertices of degree 2 that is embedded in the plane and whose leaves are then connected in a cycle determined by the embedding. 


So the OP's class of graphs (which permits vertices of degree 2) is a Halin graph with some tree-edges subdivided. 


