# Is there a name for this construction of plane graphs?

Let $$G$$ be a plane graph. Let $$G'$$ be a supergraph of $$G$$ obtained by inserting a face vertex in each face of $$G$$ and connecting the face vertex to all vertices on the boundary of the face.

For example, let $$G$$ be the left graph of the following picture. $$G'$$ is on the right. We mark the newly added vertices and edges in pink. Does the resulting planar graph from above construction have a nice name like the dual graph?

• It's analogous to barycentric subdivision. Sep 6, 2022 at 13:56
• It seems related to Apollonian Networks: en.wikipedia.org/wiki/Apollonian_network
– dbal
Sep 6, 2022 at 23:29
• @dbal yes. When $G$ is a triangulation, $G’$ is an Apollonian. Sep 7, 2022 at 0:16