Question: Find the vertex- face correspondence and edge - edge correspondence of the dual graph and the original graph.

PlaneDual returns the plane dual of a planar graph G, that is, a graph with faces of G as its vertices in which two vertices are adjacent if and only if they share an edge as faces of G. Of course, this is a little different from the standard definition of plane dual. (Interlude: I estimate that the two definitions are equivalent in the case that the planar graph is 3-connected simple graph)

It's not hard to find a plane dual of a planar graph in Maple. 

Since all labels of the dual graph are used numbers 1..n in maple, I cannot see how its vertices correspond to the face of the original graph.  And further, I want to know one edge of the dual graph corresponds to which edge (should be the boundary on two faces) of the original planar graph.

Maybe input {1,4} of the dual graph and output {2,4} of the original graph.

For example, Input {1} that is a vertex of  dual graph  to get the original face {2,3,4} and if we input {1,4} of the dual graph, we will output edge {2,4} of the original graph.

I don't know if there's a good way to do that.