"""Generator for Sudoku graphs
This module gives a generator for n-Sudoku graphs. It can be used to develop
algorithms for solving or generating Sudoku puzzles.
A completed Sudoku grid is a 9x9 array of integers between 1 and 9, with no
number appearing twice in the same row, column, or 3x3 box.
+---------+---------+---------+
| | 8 6 4 | | 3 7 1 | | 2 5 9 |
| | 3 2 5 | | 8 4 9 | | 7 6 1 |
| | 9 7 1 | | 2 6 5 | | 8 4 3 |
+---------+---------+---------+
| | 4 3 6 | | 1 9 2 | | 5 8 7 |
| | 1 9 8 | | 6 5 7 | | 4 3 2 |
| | 2 5 7 | | 4 8 3 | | 9 1 6 |
+---------+---------+---------+
| | 6 8 9 | | 7 3 4 | | 1 2 5 |
| | 7 1 3 | | 5 2 8 | | 6 9 4 |
| | 5 4 2 | | 9 1 6 | | 3 7 8 |
+---------+---------+---------+
The Sudoku graph is an undirected graph with 81 vertices, corresponding to
the cells of a Sudoku grid. It is a regular graph of degree 20. Two distinct
vertices are adjacent if and only if the corresponding cells belong to the
same row, column, or box. A completed Sudoku grid corresponds to a vertex
coloring of the Sudoku graph with nine colors.
More generally, the n-Sudoku graph is a graph with n^4 vertices, corresponding
to the cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
only if they belong to the same row, column, or n by n box.
References
----------
.. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
polynomials. Notices of the AMS, 54(6), 708-717.
.. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
.. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
"""
import networkx as nx
from networkx.exception import NetworkXError
__all__ = ["sudoku_graph"]
[docs]
@nx._dispatchable(graphs=None, returns_graph=True)
def sudoku_graph(n=3):
"""Returns the n-Sudoku graph. The default value of n is 3.
The n-Sudoku graph is a graph with n^4 vertices, corresponding to the
cells of an n^2 by n^2 grid. Two distinct vertices are adjacent if and
only if they belong to the same row, column, or n-by-n box.
Parameters
----------
n: integer
The order of the Sudoku graph, equal to the square root of the
number of rows. The default is 3.
Returns
-------
NetworkX graph
The n-Sudoku graph Sud(n).
Examples
--------
>>> G = nx.sudoku_graph()
>>> G.number_of_nodes()
81
>>> G.number_of_edges()
810
>>> sorted(G.neighbors(42))
[6, 15, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 51, 52, 53, 60, 69, 78]
>>> G = nx.sudoku_graph(2)
>>> G.number_of_nodes()
16
>>> G.number_of_edges()
56
References
----------
.. [1] Herzberg, A. M., & Murty, M. R. (2007). Sudoku squares and chromatic
polynomials. Notices of the AMS, 54(6), 708-717.
.. [2] Sander, Torsten (2009), "Sudoku graphs are integral",
Electronic Journal of Combinatorics, 16 (1): Note 25, 7pp, MR 2529816
.. [3] Wikipedia contributors. "Glossary of Sudoku." Wikipedia, The Free
Encyclopedia, 3 Dec. 2019. Web. 22 Dec. 2019.
"""
if n < 0:
raise NetworkXError("The order must be greater than or equal to zero.")
n2 = n * n
n3 = n2 * n
n4 = n3 * n
# Construct an empty graph with n^4 nodes
G = nx.empty_graph(n4)
# A Sudoku graph of order 0 or 1 has no edges
if n < 2:
return G
# Add edges for cells in the same row
for row_no in range(n2):
row_start = row_no * n2
for j in range(1, n2):
for i in range(j):
G.add_edge(row_start + i, row_start + j)
# Add edges for cells in the same column
for col_no in range(n2):
for j in range(col_no, n4, n2):
for i in range(col_no, j, n2):
G.add_edge(i, j)
# Add edges for cells in the same box
for band_no in range(n):
for stack_no in range(n):
box_start = n3 * band_no + n * stack_no
for j in range(1, n2):
for i in range(j):
u = box_start + (i % n) + n2 * (i // n)
v = box_start + (j % n) + n2 * (j // n)
G.add_edge(u, v)
return G
