This post is part of a series of articles about backtrack search in groups. You should really read intro to backtrack and search reduction 1 first. This page will discuss implementing a modern graph isomorphism tester.
Before reading this post, you should go and read about ordered partitions, or at least be aware of that post. It introduces some important data structures we are going to use here.
Before we begin, we need to represent a graph! There are many papers, which discuss how to efficiently represent graphs. There are also excellent packages in GAP which can efficiently represent graphs. For this post we are going to use the digraphs package. This already includes bliss, a high quality graph isomorphism detector, but we are going to write our own anyway!
Firstly, we need to load the digraphs package (if you don't have it, you might need a more recent GAP)
There are several ways to build a directed graph. We will use the method of giving the adjacency list – this is a list
L[i] is the vertices which
i is connected to.
d1 := Digraph([ , , , , [1,2,3,4] ]); # <digraph with 5 vertices, 8 edges> Print(d1); # Digraph( [ [ 2 ], [ 3 ], [ 4 ], [ 1 ], [ 1, 2, 3, 4 ] ] ) d2 := Digraph([ , , [4,5], [1,5],  ]); # <digraph with 5 vertices, 6 edges> Print(d2); # Digraph( [ [ 2 ], [ 3 ], [ 4, 5 ], [ 1, 5 ], [ ] ] )
So, what do we want to find? We want to find the automorphisms of a digraph – the group of permutations on the vertices of a digraph which map the digraph to itself. In GAP this is easy to calculate. The function to find the image of a digraph under a permutation is
OnDigraphs, and we can use
Filtered to look through the
SymmetricGroup for all permutations which map a graph to itself, like this!
Filtered(SymmetricGroup(5), g -> (OnDigraphs(d1, g) = d1)); # [ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) ] Filtered(SymmetricGroup(5), g -> (OnDigraphs(d2, g) = d2)); # [ () ]
So, the automorphism group of
d1 is the cyclic group on 4 points, and for
d2 it is the Trivial group. So, we are done, article finished!
Except, this is a very inefficient way of calculating the automorphism group of a digraph. Surely we can be more efficient? Actually, we can't do much better than this in the worst case. There are graphs where the best algorithms known don't perform (much) better than this. However, for many many graphs we can do much, much better. Let's look at how.
This first part reduces the amount of search we have to do by looking at the graph. Let's look at one of our graphs:
(Don't like how the graph is laid out? You can drag vertices around, or use the drop-down menu to pick another graph!)
How can we make finding automorphisms easier? Let's begin with a super-simple theorem (so simple, it barely feels like it qualifies!)
It is a truth universally acknowledged, that a vertex in possession of $n$ neighbours, must be mapped by any automorphism to another vertex with $n$ neighbours.
So, the one vertex in the graph above with 4 neighbours must be mapped to itself. Click the filter button, which will colour vertices so only vertices with the same number of in, and out, edges are the same colour.
With some graphs, we can do much better. Let's look at our other example graph:
Press filter once, and you will see we split our vertices into three classes:
So we have already reduced our search from the original $5!$ permutations in $S_5$ to $4$ permutations (we can choose if we swap the two vertices in class 1, and independently choose if we swap the two vertices in class 2).
However, we can go further! We can extend our mini-theorem. Not only must a vertex map to a vertex with the same number of in-edges and out-edges, but if a vertex has (for example) one edge which connects to a class 1 vertex, it's image must also have an edge connected to a class 1 vertex.
Press filter again to see what difference this makes. All vertices are now different colours, and we have proved this graph has only one automorphism – the trivial one. For completeness, here are the classes we have deduced:
A) Vertex from class 1: One edge going to class 1, one from class 2 B) Vertex from class 1: One edge going to class 2, one from class 1 C) Vertex from class 2: One edge going to class 2, one edge from classes 1 and 3. D) Vertex from class 2: One edge going to class 1, one edge from classes 2 and 3. E) Vertex from class 2: Two edges going to class 2.
Now, this process doesn't always work – there are graphs with only the trivial isomorphism, but where this process is unable to split the vertices into classes – for example, the Frucht graph:
There are methods of filtering the Frucht graph. For example, some vertices have two connected neighbours, while other vertices have three disconnected neighbours. For now, let's consider how we can implement the most simple kind of filtering.
filterGraph := function(cells, digraph) local celllist, filter, f, edge; celllist := CellsToList(cells); # Start making a lists of lists, whose # first member is the original cell number filter := List(celllist, x -> [x]); # Add each edge for edge in DigraphEdges(digraph) do Add(filter[edge], [ 1, celllist[edge] ]); Add(filter[edge], [-1, celllist[edge] ]); od; for f in filter do Sort(f); od; return ListToCells(filter); end;
fullyPropagateConstraints := function(cells, conlist) local cellcount, con; # Make -1 to force at least one loop to occur cellcount := -1; while cellcount <> Length(cells) do cellcount := Length(cells); for con in conlist do cells := con(cells); od; od; return cells; end;
conList := [ x -> filterGraph(x, d1), x -> filterGraph(x, d2) ];; fullyPropagateConstraints([[1..5]], conList);
branchFirstCell := cells -> First([1..Length(cells)], x -> Size(cells[x]) > 1); branchSmallCell := function(cells) local bestindex, bestsize, i; bestindex := fail; bestsize := infinity; for i in [1..Length(cells)] do if Size(cells[i]) > 1 and Size(cells[i]) < bestsize then bestindex := i; bestsize := Size(cells[i]); fi; od; return bestindex; end;
# rBase := rec( refinedCells := , branchCell := , branchValue := , nextLevel := ) buildrBase := function(cells, constraintList, branchOrder) local rBaseRoot, rBase, branchCell; rBaseRoot := rec(); rBase := rBaseRoot; while true do cells := fullyPropagateConstraints(cells, constraintList); branchCell := branchOrder(cells); rBase.cells := StructuralCopy(cells); if branchCell = fail then return rBaseRoot; fi; rBase.branchCell := branchCell; rBase.branchValue := Minimum(cells[branchCell]); cells := FixPoint(cells, rBase.branchValue); rBase.nextLevel := rec(); rBase := rBase.nextLevel; od; end;