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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.

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  • $\begingroup$ another example of those graphs is the union of a strictly convex polygon with its Voronoi diagram $\endgroup$
    – Manfred Weis
    Commented Mar 28, 2018 at 12:52
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    $\begingroup$ How about “Halin graph”? $\endgroup$
    – Gordon Royle
    Commented Mar 28, 2018 at 13:18
  • $\begingroup$ @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. $\endgroup$
    – Manfred Weis
    Commented Mar 28, 2018 at 14:01

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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.

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