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Suppose I have a hypergraph with vertices V and hyperedges H, where each hyperedge is a subset of V.

I want to form a normal graph with vertex set V, where two vertices are adjacent if they lie in the same hyperedge.

What is this graph called?

If this were a geometry with V being the points and H the lines, then I would call it either the "point graph" or the "collinearity graph", but I want to use the same concept in a more general non-geometric setting for which the name collinearity graph seems inappropriate, but "point graph" seems insufficiently descriptive.

Batominovski
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Gordon Royle
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    I've heard the graph referred to as the "underlying graph" of $H,$ but I don't know how widespread this term is. – D Poole Feb 04 '14 at 22:02

1 Answers1

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According to one of the Wikipedia editors:

The primal graph of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge. The primal graph is sometimes also known as the Gaifman graph of the hypergraph.

Snowball
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    Hmm, thanks for answering, but I don't think this is common terminology. It seems to be used mostly (or entirely?) by people modelling constraint-satisfaction problems or similar things, one of whom presumably wrote the Wikipedia entry. – Gordon Royle Feb 04 '14 at 08:04