Are there any practical situations that can be represented using a modified definition of an edge - where each edge connects three vertices instead of two?
An edge that connects 3 vertices
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9Sure, for example the notion of three points being collinear. See http://en.wikipedia.org/wiki/Hypergraph . – 2012-08-12
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0Would a triangle do? It sort of connects 3 vertices. Another possibility: in a directed graph one has two functions $s,t: E \to V$ called source and target. So one could up this to three functions! Then any selection of 2 of these would define a directed graph. – 2012-08-13
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3The relevant term is "hypergraph", which have a large literature. Edges in a hypergraph may be an arbitrary subset of the vertex set. – 2012-08-19
1 Answers
You're talking about $3$-uniform hypergraphs. They will arise in a range of contexts, some of which are:
Steiner triple systems are $3$-uniform hypergraphs in which each pair of vertices belongs to exactly one hyperedge.
Triangulations of surfaces (e.g. the plane).
The problem of completing a partial triangulation of the complete tripartite graph $K_{n,n,n}$ has been shown to be NP-complete. Charlie Colbourn gave a polynomial time reduction of this problem to the problem of completing partial Latin squares, thereby showing that this problem is also NP-complete (ref.).
These contexts are more areas of pure mathematics, and can be rephrased to not be $3$-uniform hypergraphs (however, I'm guessing all $3$-uniform hypergraph problems can be thus rephrased).
Combinatorial designs, such as Steiner triple systems, have applications in software testing. Hypergraphs in general arise in the study of complex networks, but I haven't seen the particular case of $3$-uniform hypergraphs arise.
However, we don't need applications for mathematics to be interesting; the following quote applies to mathematics as well as physics.
Physics is like sex. Sure you can get some interesting results, but that's not why we do it. -- Richard Feynman (disputed).