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If I have a graph consisting of 2 disjoint triangles, which are connected by an additional edge, then I have difficulties understanding how its dual graph looks like.

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Your graph has 3 vertices: one for each triangle and one for the infinite face. Lets call these vertices 1,2 and 3, the last being infinite. There are 3 edges separating 1,3 thus in the dual graph you get 3 edges between 1 and 3. Same with 2 and 3. Also the edge connecting 1 and 2 becomes a loop at 3 in the dual graph. –  N. S. Jan 16 '12 at 17:44

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up vote 4 down vote accepted

Here is a picture. The dual graph is in green. Here's a picture.

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Oops, why the downvote? Did I make an obvious mistake? –  Grumpy Parsnip Jan 16 '12 at 19:16
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Perhaps this is my karmic penance for wasting so much time on the internet today... –  Grumpy Parsnip Jan 16 '12 at 19:30
    
This is sometimes called the "geometric dual." Note that the original graph has no loops or multiple edges but the "dual graph" has multiple edges and one can construct examples where the "geometric dual" also has loops even if the original graph did not. –  Joseph Malkevitch Jan 16 '12 at 23:23

The dual graph consist of three vertices 1, 2 and 3 such that there are three edges between 1 and 2, three edges between 2 and 3, one loop at the vertex 2.

Note that the number of faces of the graph equal the number of vertices in the dual graph, number of edges equal the number of edges and the number of faces equal the number of vertices of the dual graph.

Further a planar connected graph is isomorphic to its double dual.

The following image is the best I could get with MS Paint. Sorry to have it hoorible!

Dual Graph

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Thanks, but sorry I don't understand how the vertices of the primal can be vertices in the dual, –  alex Jan 16 '12 at 17:03
    
I think we use different definitions of dual: The dual in my sense has a vertex for each face in the primal –  alex Jan 16 '12 at 17:23
    
I realise I have made a mistake. I shall edit in a few minutes. –  user21436 Jan 16 '12 at 17:25

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