The 
-double
cone graph is the name given in the work to the graphs that geometrically correspond
to a double cone obtained by connecting all vertices
of two 
cycle graphs (each corresponding with one nappe)
with a single additional point. The 
-double cone graph therefore has 
vertices.
The 3-double cone graph is isomorphic to the 
-windmill graph.
Double cone graphs are unit-distance and planar, as illustrated in first and third, respectively, sets of embeddings above.
