Prove two graphs are isomorphic from geometric duals.

Question

Prove that two graphs $G_1$ and $G_2$ are isomorphic if and only if their geometric duals $G_1^*$ and $G_2^*$ are also isomorphic.

Answer 1

This is false. Consider the following graphs:

Here, clearly $G \cong H$ but $G^* \ncong H^*$ since $G^*$ has a vertex of degree $5$ whereas $H^*$ does not. Note that the colors of the vertices are chosen so that they represent the color of the faces of the original graphs. White is used for the exterior face.