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Although I have some ideas on how to create a dual graph from a planar graph, I preferred to ask it here. Is there any algorithm for this purpose?

Thank you so much.

Rodrigo de Azevedo
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m.taheri
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    Keep in mind that the dual depends on the planar embedding. So it's not a property of the graph in abstract; geometry will be necessary. – Robert Mastragostino Aug 15 '13 at 11:41
  • Assume that we want to obtain the dual from a specific embedding of a graph. So what is the algorithm for this purpose? – m.taheri Aug 15 '13 at 17:01

2 Answers2

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You may have heard of the Quad-Edge data structure for graph representation (https://en.wikipedia.org/wiki/Quad-edge). It has the advantage to represent the graph and its dual simultaneously, so once the graph is built, there is no need to "dualize" anymore.

I suspect that the Winged-edge structure (or even DCEL) has the same capability but this needs to be confirmed.

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    If a planar graph has multiple embeddings, then we may have multiple dual graphs for the planar graph. Then how this structure handles this issue? – m.taheri Oct 04 '16 at 18:09
  • I imagine that you need to instantiate every embedding. –  Oct 04 '16 at 18:19
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Take the set of faces as set of vertices and add an edge $f_1f_2$ whenever an edge separates these two faces.

Hagen von Eitzen
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    And what is the algorithm to take the set of faces? – m.taheri Aug 15 '13 at 18:43
  • @m.taheri The faces should more or less be given. More precisely, we cannot compute the dual of a planar graph, but only the dual of a plane graph. As what kind of data structure is your graph and embedding into the plane given? – Hagen von Eitzen Aug 16 '13 at 14:00
  • Imagine we have a specific embedding of a planar graph. In this case i need the algorithm for a fullerene graph. A fullerene is a 3-regular planar graph with pentagonal and hexagonal faces. – m.taheri Aug 17 '13 at 18:42