3
$\begingroup$

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.

  • 1
    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.2012-01-16

2 Answers 2

6

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

  • 0
    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.2012-01-16
2

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

  • 0
    I realise I have made a mistake. I shall edit in a few minutes.2012-01-16