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Is there a name for a kind of graph where edges are vertices in the same graph ?

A example would be :

e1(a,b) e2(c,d) e3(e1,e)

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    This could pretty much be interpreted as the hypergraph with edges $\{c,d\}$ and $\{a,b,e\}$ if that is what you want.2012-10-10

2 Answers 2

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Not sure but in RDF you have that. They are called labelled directed graphs in the "RDF Concepts" spec and directed graphs in RDF Semantics.

For example in Turtle - which is just one notation for RDF Graphs - you can write

@prefix foaf: . @prefix rdf: .  rdf:type a rdf:Property .  rdf:type foaf:Person . 

Here the rdf:type name is in subject position in the first statement (a vertice?) and in predicate position (an edge) in the second sentence .

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Updated Answer

I am not sure if this is exactly what you are looking for, but one way to have the edges as nodes in the same graph is using Bipartite graphs:

http://en.wikipedia.org/wiki/Bipartite_graph

For example,

$G = (V,E) = (\{ v_1, v_2 \}, \{ (v_1,v_2) \}).$

Can be represented by,

$G' = (V',U,E') = (\{ v_1, v_2 \}, \{ (v_1,v_2) \}, \{ (v_1,(v_1,v_2)), (v_2,(v_1,v_2))\}).$

Where $(v_1,v_2)$ is now a node, and nodes $v_1$,$v_2$ is connected to it.

Previous Wrong Answer

They are sometimes called "dual graphs" (e.g. http://people.hofstra.edu/geotrans/eng/methods/dual_graph.html). And sometimes called "edge dual graphs".

However, I understand that "dual graphs" can also refer to the dual graphs of planar graphs.

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    @RahulNarain - $(v_1,v_2)$ is an edge in $G$, not a vertex. The only vertices are $v_1,v_2$. I wrote $G,G'$ in "(vertices,edges)" format. So, $G'$ actually does reflect the connectivity of $G$. But yes, we cannot meet the requirements of the poster. I was suggesting a separate hypergraph/bipartite graph $G'$ that "represents" the connectivity of $G$. Instead of $G'$ representing its own connectivity.2012-10-11