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Duals and Inner Duals of Planar Graphs

Graphs that can be embedded in the plane are called planar graphs - that is, they can be drawn without crossings. These drawings (or 'embeddings') have a dual which also corresponds to a graph, with a vertex in the dual for every face in the original, and an edge in the dual for every pair of faces that share an edge.
For example, for each of the embeddings above (in black) of the same graph there is a different dual drawing corresponding to different graphs. If the vertex that represents the outer face is deleted from a dual, you get an inner dual.

These inner duals are what Brinkmann et al use for generating fusanes, as I mentioned. The basic idea is to generate the duals, and then expand those into fusanes. Since the (inner) duals necessarily have less vertices than the full graph, this is quite efficient.

However, I admit that I couldn't be bothered to implement the algorithm properly, even though the technique is partly based on McKay's method which I know a little about. Instead, I simply generated trees for the duals. These then have to be 'labeled' - confusingly this is not the same kind of labeling as for a labeled tree. Instead, the labels are the number of edges deleted from the dual to make the inner dual.

This image shows a single tree with two possible labelings, and the fusanes that correspond to those labelings. The middle vertices of the tree are labeled with a pair (1/3, 3/1, or 2/2) because there are two distinct parts to the outside curve. It might be obvious that the graph on the left - which corresponds to a labeling of 5,1/3,3/1,5 - would be identical to the one made by a labeling of 5,3/1,1/3,5. Or, to put it another way, you can rotate it through 180 degrees and get the same graph.

Here are all the fusanes made by this technique from tree-inner-duals on 7 vertices:

the far-left in the center is not a cycle, it's just the branches meet. There's also a duplicate in there!

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