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?