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)
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)
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 .
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.