## Practical Graph Isomorphism

the individualization-refinement method

### Isomorphism and Automorphism

We have some finite objects consisting of some finite sets
and some relations on them.
An *isomorphism* between two objects is a bijection
between their sets so that the relations are preserved.
Isomorphism is an equivalence relation on the set of all objects.
Isomorphisms from an object to itself are *automorphisms*.
The set of automorphisms forms a group under composition.
It is called the *automorphism group*.

(1)

(1 4) (2 3) (5 8) (6 7)

(1 8) (2 7) (3 6) (4 5)

(1 5) (2 6) (3 7) (4 8)

The *graph isomorphism problem* is: (GI)
*Given two graphs, determine whether they are isomorphic*.
The fastest known algorithm for GI is
*quasipolynomial*;
no polynomial time algorithm is known (despite many claims).
GI is not known to be NP-complete. GI is not known to be in co-NP.
There are polynomial time algorithms for many special classes of graphs
(bounded degree, bounded genus, classes with excluded minors or topological
subgraphs). Few of these are practical, planar graphs being a notable exception.
It is possible that group isomorphism is easier and that permutation
equivalence of linear codes is harder than GI. But nobody can prove it.

#### Ubiquity of Graph Isomorphism

Very many isomorphism problems can be efficiently expressed as graph isomorphism
problems. For convenience, we use coloured graphs, in which the vertices (and possibly
the edges) are coloured. Isomorphisms must preserve vertex and edge colour.

# Matrix Isomorphism

# Hadamard Equivalence

# Isotopy

# Group Isomorphism

Given a class of objects, and a definition of isomorphism,
we can choose one member of each isomorphism class and call it the
*canonical* member of the class.
The process of finding the canonical member of the isomorphism class containing
a given object is called *canonical labeling*.
Two labeled objects which are isomorphic become identical when they are
canonically labeled.
Since identity of objects is usually far easier to check than isomorphism,
canonical labeling is the preferred method of sorting a collection of objects
into isomorphism classes.
A possible definition of the canonical member of an isomorphism class would be
the member which maximises some linear representation (such as a list of edges).
In practice, most programs compute a canonical member which is easy for
computers to find rather than easy for humans to define.

### Refinement: equitable partition

A key concept used by **nauty** and other tools is
*partition refinement* (partition = colouring).
An *equitable* partition is one where every two vertices of the same colour are
adjacent to the same number of vertices of each colour.

Equitable partitions

Refinement of a partition means to subdivide its cells.
Given a partition (colouring) π, there is a unique equitable partition
that is a refinement of π and has the least number of colours.
The next picture shows the steps which lead from an initial colouring to
the equitable partition. The black arrows indicate vertices having a
number of neighbours in a reference cell which is different from that of other
vertices in their cell.

Refinement steps

All competitive isomorphism programs generate a tree based on
partition refinement. The nodes of the tree correspond to partitions (colourings).
The root of the tree corresponds to the initial colouring, refined.
If a node corresponds to a discrete partition
(each vertex with a different colour), it has no children and is a leaf.
Otherwise we choose a colour used more than once (the target cell),
individualize one of those vertices by giving it a new unique colour, and refine
to get a child.
Each leaf lists the vertices in some order (since colours have a predefined order),
so it corresponds to a labelling of the graph.
If we have two leaves giving the same labelled graph,
we have found an automorphism.
If we define an order on labelled graphs, such as lexicographic order,
then the greatest labelled graph corresponding to a leaf is a canonical graph.
However the tree can be much too large to generate completely.
It will be pruned by means of *node invariants*.

### Node invariants and pruning

A

*node invariant* is a value Φ(ν) attached to each node ν
in the tree that depends only on the combinatorial properties of ν
and its ancestors in the search tree (not on the labelling of the graph).
Node invariants, together with automorphisms, allow us to remove parts of the search
tree without generating them.

- If ν1,
ν2
are nodes at the same level in the search tree and

Φ(ν1) <
Φ(ν2), then
no canonical labelling is descended from ν1.
- If ν1,
ν2
are nodes at the same level in the search tree and

Φ(ν1) ≠
Φ(ν2), then
no labelled graph descended from ν1 is the same as one descended from
ν2.
- If ν1,
ν2
are nodes with ν2 =
(ν1)
*g*
for an automorphism *g*, then *g* maps the subtree
descended from ν1
onto the subtree descended from ν2.

Each of these observations enables us to prune parts of the search tree.
The hope is that the remaining parts will be small enough to compute.

Search tree without pruning

### Equitable Partition and Orbit Partition - An Example

Equitable partitions are in general unable to capture the *orbit partition* of vertices of
a graph, as shown by the next example: vertices 3, 4, 5 and 6 are in the same cell of the equitable partition but there isn't any automorphism between vertices in {3, 4} and in {5, 6}.

# Initial graph

# Count neighbours of green vertices

# Equitable partition

# Orbit partition

The individualization-refinement method allows − in two steps −
for detecting an automorphism which maps vertex 3 into 5.

Remark: Assume that an ordered sequence of *n*
(the number of vertices of the graph) colours is given.
The individualized vertex keeps the colour of the cell to which
it belongs, while the remaining vertices in that cell are assigned the next colour.

# Individualize vertex 3 (green) and count its neighbours

# Count neighbours of blue vertices

# Equitable partition

# Individualize vertex 5 (cyan) and count its neighbours

# Equitable (and discrete) partition

# Individualize vertex 4 (green) and count its neighbours

# Count neighbours of blue vertices

# Equitable partition

# Individualize vertex 5 (cyan) and count its neighbours

# Equitable (and discrete) partition

The automorphism (3 4) (7 9) (8 10)
is found by comparing vertices of the coloured graphs (D) and (D').

### Base and strong generating set - Group size

Example of a base and strong generating set. For a graph with *n* vertices there are at most *n − 1* generators, even though the group size can be up to *n!*

# 1) Fixing 1,5,9,13

# 2) Fixing 1,5,9. Orbit of 13 is {13, 15}

# 3) Fixing 1,5. Orbit of 9 is {9, 11}

# 4) Fixing 1. Orbit of 5 is {5, 7, 13, 15}

# 5) Fixing nothing.

Orbit of 1 is {1, 3, 5, 7, 9, 11, 13, 15}

1) Everything fixed.

2) Orbit of 13 is {13, 15} (orbit size=2)

Generator: (13 15)

3) Orbit of 9 is {9, 11} (2)

(9 11)

4) Orbit of 5 is {5, 7, 13, 15} (4)

(5 7),
(2 4)(5 13)(6 16)(7 15)(8 14)(10 12)

5) Orbit of 1 is {1, 3, 5, 7, 9, 11, 13, 15} (8)

(1 3),
(1 5)(2 8)(3 7)(4 6)(9 13)(10 16)(11 15)(12 14)

Group size = 2 × 2 × 4 × 8 = 128.