How do you represent a graph in Haskell?

Question

It\'s easy enough to represent a tree or list in haskell using algebraic data types. But how would you go about typographically representing a graph? It seems that you need

Answer 1

It's true, graphs are not algebraic. To deal with this problem, you have a couple of options:

1. Instead of graphs, consider infinite trees. Represent cycles in the graph as their infinite unfoldings. In some cases, you may use the trick known as "tying the knot" (explained well in some of the other answers here) to even represent these infinite trees in finite space by creating a cycle in the heap; however, you will not be able to observe or detect these cycles from within Haskell, which makes a variety of graph operations difficult or impossible.
2. There are a variety of graph algebras available in the literature. The one that comes to mind first is the collection of graph constructors described in section two of Bidirectionalizing Graph Transformations. The usual property guaranteed by these algebras is that any graph can be represented algebraically; however, critically, many graphs will not have a canonical representation. So checking equality structurally isn't enough; doing it correctly boils down to finding graph isomorphism -- known to be something of a hard problem.
3. Give up on algebraic datatypes; explicitly represent node identity by giving them each unique values (say, Ints) and referring to them indirectly rather than algebraically. This can be made significantly more convenient by making the type abstract and providing an interface that juggles the indirection for you. This is the approach taken by, e.g., fgl and other practical graph libraries on Hackage.
4. Come up with a brand new approach that fits your use case exactly. This is a very difficult thing to do. =)

So there are pros and cons to each of the above choices. Pick the one that seems best for you.