# 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 =  * 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!