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[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)