# -*- coding: utf-8 -*-

"""

Minimum rank

This module implements a collection of bounds for minimum rank, plus a

driver function to apply each of these in turn to a graph.

This code was initially published in http://arxiv.org/abs/0812.1616 by

Laura DeLoss, Jason Grout, Tracy McKay, Jason Smith, and Geoff Tims.

"""

#######################################################################

#

# Copyright (C) 2008, 2009 Laura DeLoss, Jason Grout, Tracy McKay,

# Jason Smith, and Geoff Tims.

#

#

# This program is free software: you can redistribute it and/or modify

# it under the terms of the GNU General Public License as published by

# the Free Software Foundation, either version 2 of the License, or

# (at your option) any later version.

#

# This program is distributed in the hope that it will be useful, but

# WITHOUT ANY WARRANTY; without even the implied warranty of

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU

# General Public License for more details.

#

# You should have received a copy of the GNU General Public License

# along with this program. If not, see http://www.gnu.org/licenses/.

#######################################################################

##########################################################

# Imports all the graphs of order 7 or less and stores #

# them in a list called atlas_graphs so that #

# atlas_graphs[i] is the ith graph in the atlas of #

# graphs #

##########################################################

from sage.graphs.graph import Graph

import networkx.generators.atlas

atlas_graphs = [Graph(i) for i in \

networkx.generators.atlas.graph_atlas_g()]

##########################################################

# A list "database" of all the minimum ranks of graphs #

# of order 7 or less. #

# #

# The minimum ranks are stored as ordered pairs: the #

# first coordinate is the graph number, the second #

# coordinate is the minimum rank. The first tuple in #

# the list min_ranks is just a position holder. #

##########################################################

min_ranks = [(0,None), (1,0), (2,0), (3,1), (4,0), (5,1), (6,2),

(1252,1)]

###########################################################

# The atlas of graphs is ordered by number of vertices, then number of

# edges. To make our search for minimum ranks more efficient, we

# store the first and last indices of the graphs having a certain

# number of vertices and edges.

#

# graph_help[n][m] is a tuple (i,j), meaning that the graphs with n

# vertices and m edges start at index i and end on index j in the

# atlas of graphs.

###########################################################

graph_help = {

21: (1252,1252)}}

def get_mr_from_list(graph):

raise ValueError, "This should never happen!"

def zerosgame(graph, initial_set=[]):

return list(zero_set)

def zero_forcing_set_bruteforce(graph, bound=None):

return False

def find_Z(graph):

return len(zero_forcing_set_bruteforce(graph))

def has_forbidden_induced_subgraph(graph):

return False

# if in library mode, we need to import this.

# if it is just included in a sage notebook, then it is in the global namespace

# so we try importing it if we can. If we can't import it, then just trust that

# everything is in the global namespace.

try:

from zero_forcing_set_wavefront import zero_forcing_set_wavefront

except ImportError:

pass

def min_rank_by_bounds(graph, tests = ['precomputed', 'order', 'zero forcing', 'not path', 'forbidden minrank 2', 'not planar', 'not outer planar', 'clique cover', 'diameter']):

return (lower_bound, upper_bound)

def find_rank_spread(vertex, graph):

return False,False

def cut_vertex_connected_graph_mr(c_vertex,graph):

return subgraph_mr_sum+rank_spread, subgraph_mr_sum+rank_spread

def minrank_bounds(graph, all_bounds=False, tests=['precomputed', 'order', 'zero forcing', 'zero forcing fast', 'not path', 'forbidden minrank 2', 'not planar', 'not outer planar', 'clique cover', 'cut vertex', 'disconnected', 'diameter']):

return max(lower_bound.values()), min(upper_bound.values())

# From the patch for generic_graph.py

def is_outerplanar(self, **kwds):

return h.is_circular_planar(**kwds)

# From the patch for generic_graph.py

def cut_vertex_balanced(self):

return best_v[0]

# From the patch for graph.py

def cliques_containing_edge(self, edge):

return None

# From the patch for graph.py

def edge_clique_cover_minimum(self, bound=None):

return None