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Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

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Source code for networkx.generators.sudoku

"""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]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(0, 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(0, 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