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Mathematics rating:	Start Class	Mid Priority	Field: Discrete mathematics
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 22:20, 10 June 2007 (UTC)

Contents 1 non-isomorphic 1.1 introduction - come in pairs? 1.2 example 1.3 Regular Polyhedra

[edit] non-isomorphic

...has order 6, not 7 right?... =The dual of a "graph" is not always a "graph" given that the Wikipedia has chosen the definition of "graph" to be the one that excludes loops and parallel edges (what is being called a "simple graph" in many books) the dual of a graph is not always a graph. Every connected component of a graph has to be 3-edge connected for its dual to be a graph. A cut edge corresponds to a loop in the dual, and an edge cut of two corresponds to parallel edges in the dual.

Also the dual of the dual is not the original graph if the original is disconnected. (See Bondy+Murty/Graph Theory and Applications.

Wouldn't it be better to speak about multigraphs rather than graphs to introduce dual. If we restrict ourselves to simple loopless graphs, the figure for non-isomorphic duals is wrong (parallel edges). —The preceding unsigned comment was added by Taxipom (talk • contribs) 00:18, 22 March 2007 (UTC). pom 00:18, 22 March 2007 (UTC) (sorry I forgot to sign) Also, the Whitney criterion will also be false. It seems that there is a strong evidence that duality should be introduced for multigraphs with loops allowed. pom 00:28, 22 March 2007 (UTC)

I don't know what means "the dual of a graph is not always a graph", even if we restrict the word graph to "simple graph". Although the definition is not so precise, the term "dual graph" means that it is a graph. Of course, some plane graph won't have a dual graph. Also, it is sometimes written "the dual" in this article, instead of "a dual graph". pom 00:25, 22 March 2007 (UTC)

The concept requires both multiple edges and loops. That is how it is nearly always done in technical places. McKay 06:23, 23 March 2007 (UTC)

It appears (to my untrained eye) that the figure for non-isomorphhic duals is in error. It is supposed to illustrate how "the same graph can have non-isomorphic dual graphs." However, the graph G is not identical in the top and bottom of the figure (the left-most vertex has been shifted to the interior). This is confusing to me. Kmote (talk) 20:03, 13 February 2008 (UTC)

It is topologically equivalent. There is a one-to-one matching of the points in top-G and bottom-G such that if a point A is connected to a point B in one, then A's equivalent connects to B's equivalent in the other.76.240.199.90 (talk) 01:37, 17 February 2008 (UTC)

[edit] introduction - come in pairs?

it is misleading to say they come in pairs: a graph G can have two duals G' and G" depending on its plane drawing - as explained later. —The preceding unsigned comment was added by Tejas81 (talk • contribs) 19:54, 9 May 2007 (UTC).

[edit] example

what about giving the dualism between the Delaunay Triangulation and Voronoi Diagrams as an example?

[edit] Regular Polyhedra

Should his page take note that the planar projections of the regular polyhedra come in dual pairs (tetrahedron-tetrahedron, hexahedron-octahedron, and dodecahedron-icosahedron)? —Preceding unsigned comment added by 76.252.3.170 (talk) 23:00, 1 February 2008 (UTC)