TITLE: Does the notion of graphs with vertex multiplicity exist?
QUESTION [11 upvotes]: I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have.
It is actually a way to write in a compact way a graph which has a lot of twin vertices: two vertices $u$ and $v$ are twin if $N(u)=N(v)$ (they are not adjacent).
Since the twin relation is an equivalence relation, there is a unique way to write any graph in a reduced form as a graph with vertex multiplicities.
Here is an example:

This representation is particularly useful when vertices represent some items with large quantity.
Does someone know if such a concept already exists ? What's the usual name ? Do you have some references ?

REPLY [3 votes]: This notion is complementary to the notion of vertex multiplication, in which every vertex is replaced with a homogeneous clique.  This goes back to Lovasz' proof of the Weak Perfect Graph Theorem, and probably even earlier.

REPLY [2 votes]: As well as the use of this by Lovász for perfect graphs (mentioned by earlier answers), this has been used by Häggkvist to find high-degree four-chromatic triangle-free graphs. See
Häggkvist, R. (1981), "Odd cycles of specified length in nonbipartite graphs", Graph Theory (Cambridge, 1981), pp. 89–99, MR0671908.