Example script to use Nauty from Python

Nauty can be used to reduce any graph into a normal form. In this notebook, we show how to use the Nauty functionality from Python.

import numpy as np
import oapackage

Define a function to invert a permutation.

def inverse_permutation(perm):
    inverse = [0] * len(perm)
    for i, p in enumerate(perm):
        inverse[p] = i
    return inverse

We define a graph with 5 nodes. The graph is defined by the incidence matrix of size \(5\times 5\) and a coloring with two colors.

graph = np.zeros((5, 5), dtype=int)
graph[0, 1] = graph[0, 2] = graph[0, 3] = graph[1, 3] = 1
graph = np.maximum(graph, graph.T)  # make array symmetric
colors = [0, 0, 0, 1, 1]

Reduce the graph to normal form using Nauty.

help(oapackage.reduceGraphNauty)

Help on function reduceGraphNauty in module oalib:

reduceGraphNauty(G, colors=None, verbose=1)
    Return vertex transformation reducing array to normal form

    The reduction is calculated using `Nauty <http://users.cecs.anu.edu.au/~bdm/nauty/>`_

    Args:
        G (numpy array or array_link) :   the graph in incidence matrix form
        colors (list or None): an optional vertex coloring
    Returns:
        v: relabelling of the vertices

def reduce(graph, colors):
    tr = oapackage.reduceGraphNauty(graph, colors=colors, verbose=0)
    tri = inverse_permutation(tr)

    graph_reduced = oapackage.transformGraphMatrix(graph, tri)
    colors_reduced = [colors[idx] for idx in tr]
    return graph_reduced, colors_reduced, tr


graph_reduced, colors_reduced, tr = reduce(graph, colors)

print("input graph: ")
print(graph)

print("normal form reduction: %s" % (tr,))
print("reduced graph: ")
print(graph_reduced)
print("colors reduced: %s" % (colors_reduced,))

input graph:
[[0 1 1 1 0]
 [1 0 0 1 0]
 [1 0 0 0 0]
 [1 1 0 0 0]
 [0 0 0 0 0]]
normal form reduction: (1, 2, 0, 4, 3)
reduced graph:
[[0 0 1 0 1]
 [0 0 1 0 0]
 [1 1 0 0 1]
 [0 0 0 0 0]
 [1 0 1 0 0]]
colors reduced: [0, 0, 0, 1, 1]

Apply a random permutation to the graph and reduce the graph again.

perm = np.random.permutation(5)
iperm = inverse_permutation(perm)
print("random permutation: %s" % (perm,))
graph2 = graph[perm, :][:, perm]
colors2 = [colors[idx] for idx in perm]

random permutation: [3 2 0 4 1]

Show the transformed matrix and color vector.

print(graph2)
print(colors2)

[[0 0 0 0 0]
 [0 0 0 1 1]
 [0 0 0 0 1]
 [0 1 0 0 1]
 [0 1 1 1 0]]
[1, 1, 0, 0, 0]

graph2_reduced, colors2_reduced, tr2 = reduce(graph2, colors2)

print("input graph: ")
print(graph2)

print("tr2: %s" % (tr2,))
print("reduced graph: ")
print(graph2_reduced)

print("colors2_reduced: %s" % (colors2_reduced,))

input graph:
[[0 0 1 0 1]
 [0 0 1 0 0]
 [1 1 0 0 1]
 [0 0 0 0 0]
 [1 0 1 0 0]]
tr2: (4, 1, 2, 3, 0)
reduced graph:
[[0 0 1 0 1]
 [0 0 1 0 0]
 [1 1 0 0 1]
 [0 0 0 0 0]
 [1 0 1 0 0]]
colors2_reduced: [0, 0, 0, 1, 1]

Check that the two reduced graphs are equal.

if np.all(graph_reduced == graph2_reduced):
    print("reduced arrays are equal!")
if np.all(colors_reduced == colors2_reduced):
    print("reduced colors are equal!")

reduced arrays are equal!
reduced colors are equal!