TITLE: How can I produce 'canonical' forms for rooted bipartite graphs?
QUESTION [7 upvotes]: The graphs I'm interested in are bipartite graphs with a specified root vertex. Because there's a root, all the vertices are 'graded' by their distance from the root. Because the graph is bipartite, vertices at depth $d$ are only ever connected to other vertices at depths $d \pm 1$ (and in particular not depth $d$).
When I represent these graphs, I order the vertices at each depth, and record the edges by a series of matrices, essentially the list of adjacency matrices from each depth to the next. (That is, the full adjacency matrix is symmetric and block tridiagonal, with zero diagonal blocks. I just write down the superdiagonal blocks.)
Now, if I reorder the vertices at some depth (this just permutes the rows of one matrix and the columns of the next), obviously I have the same underlying graph. I'd like an algorithm that picks a particular ordering at each depth, for each such graph, producing a 'canonical form', with the following properties:

the algorithm is idempotent; applying it a second time does nothing,
the algorithm is stable, in the sense that if you just look at the first $d$ depths of a graph, and see that that graph is already 'in canonical form', then when you produce a canonical form for the whole graph those first $d$ depths aren't changed, and
as many isomorphic graphs as possible are identified!

It may not be possible to satisfy 3. completely; for example the identity operation satisfies 1. and 2., but does a very bad job at 3. It's not essential for my application that every isomorphic pair of graphs are identified. (I'd be using this algorithm to speed up a combinatorial search of certain types of graphs, where I know that I'm unnecessarily producing many isomorphic copies of the same graph, but the details of the search require that I use this representation.)
Does anyone know of such an algorithm? Can anyone suggest something good?

REPLY [10 votes]: This problem is algorithmically equivalent to the general problem of finding a canonical labelling for a graph. To see that, take an arbitrary graph, add a new vertex adjacent to everything and call it the root, then subdivide every edge with a new vertex.  The result is a rooted bipartite graph. Going back to the original is easy.  Canonical labelling has unknown worst case complexity, though there are effective programs like nauty, bliss and Traces that can handle graphs with thousands of vertices.