I'm interested in the relationships between graphs and their duals, so I came up with the idea of a "dual meta-graph".
Given some set $S=\left\{G_0,G_1,G_2\dots\right\}$ of planar graphs, we can define the dual meta-graph of $S$ as the graph where each vertex $v$ uniquely represents one of the $G_i$ and two vertices $u$ and $v$ are connected by an edge if and only if there exists an embedding of graph $u$ (in the plane) whose dual is isomorphic to $v$.
For example, the dual-metagraph of $\left\{K_4\right\}$ is a graph containing just one vertex and one loop.
Given the dual operation $^*$, we can see that a meta-graph would be undirected since $\left(G^*\right)^*=G$ for any graph embedding G. We also know that $F(G^*)=V(G)$ and $V(G^*)=F(G)$ for any graph G, so each connected subgraph is either bipartite or has $F(G)=V(G)$ for all $G\in S$.
Does such a thing exist and, if so, what other properties is it known to have? Additionally, how can we determine its structure?