Graph Isomorphism

Question

Is there an algorithm or heuristics for graph isomorphism?

Corollary: A graph can be represented in different different drawings.

What s the best approach to

Answer 1

My project - Griso - at sf.net: http://sourceforge.net/projects/griso/ with this description:
Griso is a graph isomorphism testing utility written in C++. It is based on my own POLYNOMIAL-TIME (in this point the salt of the project) algorithm. See Griso's sample input/output on http://funkybee.narod.ru/graphs.htm page.

Answer 2

It is a hell of a problem.

In general, the basic idea is to simplify the graph into a canonical form, and then perform comparison of canonical forms. Spanning trees are generated with this objective, but spanning trees are not unique, so you need to have a canonical way to create them.

After you have canonical forms, you can perform isomorphism comparison (relatively) easy, but that's just the start, since non-isomorphic graphs can have the same spanning tree. (e.g. think about a spanning tree T and a single addition of an edge to it to create T'. These two graphs are not isomorph, but they have the same spanning tree).

Other techniques involve comparing descriptors (e.g. number of nodes, number of edges), which can produce false positive in general.

I suggest you to start with the wiki page about the graph isomorphism problem. I also have a book to suggest: "Graph Theory and its applications". It's a tome, but worth every page.

As from you corollary, every possible spatial distribution of a given graph's vertexes is an isomorph. So two isomorph graphs have the same topology and they are, in the end, the same graph, from the topological point of view. Another matter is, for example, to find those isomorph structures enjoying particular properties (e.g. with non crossing edges, if exists), and that depends on the properties you want.

Answer 3

A very similar problem - graph automorphism - can be solved by saucy, which is available in source code. This finds all symmetries of a graph. If you have two graphs, join them into one and any isomorphism can be discovered as an automorphism of the join.

Disclaimer: I am one of co-authors of saucy.

Answer 4

• nauty and Traces

nauty and Traces are programs for computing automorphism groups of graphs and digraphs [*]. They can also produce a canonical label. They are written in a portable subset of C, and run on a considerable number of different systems.

• AutGroupGraph command in GRAPE's package of GAP.
• bliss: another symmetry and canonical labeling program.
• conauto: a graph ismorphism package.

Answer 5

There are algorithms to do this -- however, I have not had cause to seriously investigate them as of yet. I believe Donald Knuth is either writing or has written on this subject in his Art of Computing series during his second pass at (re)writing them.

As for a simple way to do something that might work in practice on small graphs, I would recommend counting degrees, then for each vertex, also note the set of degrees for those vertexes that are adjacent. This will then give you a set of potential vertex isomorphisms for each point. Then just try all those (via brute force, but choosing the vertexes in increasing order of potential vertex isomorphism sets) from this restricted set. Intuitively, most graph isomorphism can be practically computed this way, though clearly there would be degenerate cases that might take a long time.

Answer 6

I recently came across the following paper : http://arxiv.org/abs/0711.2010 This paper proposes "A Polynomial Time Algorithm for Graph Isomorphism"