# Name for Biconnected Tree+Cycle Graph

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.

• another example of those graphs is the union of a strictly convex polygon with its Voronoi diagram – Manfred Weis Mar 28 '18 at 12:52
• How about “Halin graph”? – Gordon Royle Mar 28 '18 at 13:18
• @GordonRoyle yes, that fits; Wolfram makes the restriction, that there be no nodes of degree 2, but that isn't an essential restriction. It would be an acceptable answer to my question. – Manfred Weis Mar 28 '18 at 14:01

## 1 Answer

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.