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# Copyright 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Hyde
#
# This file is shared by a couple of distributions.
#
# This file 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 3, or (at your option) any later
# version.
#
# This file 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 file. If not, see <http://www.gnu.org/licenses/>.
# (Some tests in gmaker xt/MyGraphs-various.t)
package MyGraphs;
use 5.010;
use strict;
use warnings;
use Carp 'croak';
use List::Util 'min','max','sum','minstr';
use Scalar::Util 'blessed';
use File::Spec;
use File::HomeDir;
use Math::Trig ();
use POSIX 'ceil';
my @ipc;
use base 'Exporter';
use vars '@EXPORT_OK';
@EXPORT_OK = ('Graph_Easy_view',
'Graph_Easy_edges_string',
'Graph_Easy_edge_list_string',
'edge_aref_to_Graph_Easy',
'Graph_Easy_line_graph',
'Graph_Easy_print_adjacency_matrix',
'edge_aref_to_Graph',
'Graph_view',
'Graph_tree_print','Graph_xy_print',
'Graph_print_tikz',
'Graph_branch_reduce',
'Graph_is_regular',
'Graph_is_isomorphic',
'Graph_is_subgraph',
'Graph_is_induced_subgraph',
'Graph_from_edge_aref',
'Graph_line_graph',
'Graph_is_line_graph_by_Beineke',
'Graph_Wiener_index','Graph_Wiener_part_at_vertex',
'Graph_terminal_Wiener_index','Graph_terminal_Wiener_part_at_vertex',
'Graph_to_sparse6_str',
'Graph_to_graph6_str',
'Graph_from_graph6_str',
'Graph_triangle_is_even',
'Graph_triangle_search','Graph_find_triangle',
'Graph_has_triangle','Graph_triangle_count',
'Graph_claw_search',
'Graph_has_claw','Graph_claw_count','Graph_claw_count',
'Graph_clique_number',
'Graph_width_list',
'Graph_is_cycles',
'Graph_find_all_cycles',
'Graph_find_all_4cycles',
'Graph_is_hanging_cycle',
'Graph_delete_hanging_cycles',
'Graph_girth', # smallest cycle
'Graph_is_Hamiltonian',
'Graph_rename_vertex','Graph_pad_degree',
'Graph_eccentricity_path',
'Graph_tree_centre_vertices',
'Graph_tree_domnum',
'Graph_tree_domsets_count','Graph_tree_minimal_domsets_count',
'Graph_is_domset','Graph_is_minimal_domset',
'Graph_domset_is_minimal',
'Graph_minimal_domsets_count_by_pred',
'Graph_is_total_domset',
'edge_aref_num_vertices',
'edge_aref_is_subgraph',
'edge_aref_is_induced_subgraph',
'edge_aref_degrees_allow_subgraph',
'edge_aref_string',
'edge_aref_to_parent_aref',
'edge_aref_degrees',
'edge_aref_degrees_distinct',
'edge_aref_is_regular',
'parent_aref_to_edge_aref',
'parent_aref_to_Graph_Easy',
'graph6_str_to_canonical',
'graph6_view',
'make_tree_iterator_edge_aref',
'make_graph_iterator_edge_aref',
'hog_searches_html','hog_grep',
'postscript_view_file',
'Graph_to_GraphViz2',
'Graph_set_xy_points',
'Graph_subtree_depth',
'Graph_subtree_children',
'Graph_star_replacement','Graph_cycle_replacement',
);
# uncomment this to run the ### lines
# use Smart::Comments;
#------------------------------------------------------------------------------
# Graph::Easy extras
# $filename is a postscript file
# synchronous => 1, wait for viewer to exit before returning.
sub postscript_view_file {
my ($filename, %options) = @_;
require IPC::Run;
my @command = ('gv',
'--scale=.7',
$filename);
if ($options{'synchronous'}) {
IPC::Run::run(\@command);
} else {
push @ipc, IPC::Run::start(\@command,'&');
}
}
END {
foreach my $h (@ipc) {
$h->finish;
}
}
# $graph is a Graph::Easy object, show it graphically
# synchronous => 1, wait for viewer to exit before returning.
sub Graph_Easy_view {
my ($graph, %options) = @_;
require File::Temp;
my $dot = File::Temp->new (UNLINK => 0, SUFFIX => '.dot');
my $dot_filename = $dot->filename;
# per Graph::Easy::As_graphviz
print $dot $graph->as_graphviz;
graphviz_view_file($dot_filename, %options);
}
# $str is DOT format graph
sub graphviz_view {
my ($str) = @_;
graphviz_view_file(\$str);
}
# $filename is a filename string or a scalar ref to string contents
sub graphviz_view_file {
my ($filename, %options) = @_;
require File::Temp;
my $ps = File::Temp->new (UNLINK => 0, SUFFIX => '.ps');
my $ps_filename = $ps->filename;
### $ps_filename
require IPC::Run;
IPC::Run::run(['dot','-Tps',
],
'<',$filename, '>',$ps_filename);
# ['neato','-Tps','-s2']
postscript_view_file ($ps->filename, %options);
}
sub Graph_Easy_branch_reduce {
my ($graph) = @_;
foreach my $node ($graph->nodes) {
my @predecessors = $node->predecessors();
my @successors = $node->successors();
if (@predecessors == 1 && @successors == 1) {
$graph->del_node($node);
$graph->add_edge($predecessors[0], $successors[0]);
}
}
}
sub Graph_Easy_leaf_reduce {
my ($graph) = @_;
# print "$graph";
foreach my $node ($graph->nodes) {
my @successors = $node->successors;
@successors == 2 || next;
if (Graph_Easy_Node_is_leaf($successors[0])
&& Graph_Easy_Node_is_leaf($successors[1])) {
$graph->del_node($successors[1]);
}
}
}
sub Graph_Easy_Node_is_leaf {
my ($node) = @_;
my @successors = $node->successors;
return (@successors == 0);
}
sub Graph_Easy_edges_string {
my ($easy) = @_;
Graph_Easy_edge_list_string($easy->edges);
}
sub Graph_Easy_edge_list_string {
my @edges = map { [ $_->from->name, $_->to->name ] } @_;
@edges = sort { $a->[0] cmp $b->[0] || $a->[1] cmp $b->[1] } @edges;
return join(' ',map {join('-',@$_)} @edges);
}
sub Graph_Easy_print_adjacency_matrix {
my ($easy) = @_;
my $has_edge_either = ($easy->is_directed
? \&Graph::Easy::As_graph6::_has_edge_either_directed
: 'has_edge');
my @vertices = $easy->sorted_nodes('name');
foreach my $from (0 .. $#vertices) {
foreach my $to (0 .. $#vertices) {
print $easy->$has_edge_either($vertices[$from], $vertices[$to]) ? ' 1' : ' 0';
}
print "\n";
}
}
#------------------------------------------------------------------------------
# Graph.pm extras
sub Graph_sorted_vertices {
my ($graph) = @_;
my @vertices = $graph->vertices;
my $func = cmp_func(@vertices);
return sort $func @vertices;
}
sub Graph_print_adjacency_matrix {
my ($graph, $fh) = @_;
$fh //= \*STDOUT;
my @vertices = Graph_sorted_vertices($graph);
print $fh "[" or die;
foreach my $from (0 .. $#vertices) {
foreach my $to (0 .. $#vertices) {
print$fh $graph->has_edge($vertices[$from], $vertices[$to]) ? '1' : '0',
$to==$#vertices ? ($from==$#vertices ? ']' : ';') : ','
or die;
}
# print "\n";
}
}
sub Graph_loopcount {
my ($graph) = @_;
my $loopcount = 0;
foreach my $edge ($graph->edges) {
$loopcount += ($edge->[0] eq $edge->[1]);
}
return $loopcount;
}
# modify $graph to branch reduce, meaning all degree-2 vertices are
# contracted out by deleting and joining their neighbours with an edge
sub Graph_branch_reduce {
my ($graph) = @_;
### Graph_branch_reduce() ...
my $more;
do {
$more = 0;
foreach my $v ($graph->vertices) {
my @neighbours = $graph->neighbours($v);
if (@neighbours == 2) {
### delete: $v
$graph->delete_vertex($v);
$graph->add_edge($neighbours[0], $neighbours[1]);
$more = 1;
}
}
} while ($more);
}
# return a list of the immediate children of vertex $v
# children are vertices compared numerically $child >= $v
sub Graph_vertex_children {
my ($graph, $v, %options) = @_;
my $cmp = $options{'cmp'} || cmp_func($graph->vertices);
my @children = grep {$cmp->($_,$v)>0} $graph->neighbours($v);
return @children;
}
sub Graph_vertex_num_children {
my ($graph, $v, %options) = @_;
my @children = Graph_vertex_children($graph,$v, %options);
return scalar(@children);
}
# $graph is a Graph.pm, show it graphically
# synchronous => 1, wait for viewer to exit before returning.
# xy => 1, treat each vertex name as x,y coordinates
sub Graph_view {
my ($graph, %options) = @_;
### Graph_view(): %options
my @vertices = $graph->vertices;
my $is_xy = ($options{'is_xy'}
|| $graph->get_graph_attribute('vertex_name_type_xy')
|| $graph->get_graph_attribute('vertex_name_type_xy_triangular')
|| $graph->get_graph_attribute('xy')
|| (@vertices
&& defined $graph->get_vertex_attribute($vertices[0],'xy'))
|| do {
my $type = $graph->get_graph_attribute('vertex_name_type');
defined $type && $type =~ /^xy/ }
|| do {
my ($v) = $graph->vertices;
defined $v && defined($graph->get_vertex_attribute($v,'x')) });
### $is_xy
if ($is_xy) {
my $graphviz2 = Graph_to_GraphViz2($graph, %options);
GraphViz2_view($graphviz2, driver=>'neato', %options);
return;
}
{
my $graphviz2 = Graph_to_GraphViz2($graph, %options);
GraphViz2_view($graphviz2, %options);
return;
}
### Convert ...
require Graph::Convert;
my $easy = Graph::Convert->as_graph_easy($graph);
### flow: $graph->get_graph_attribute('flow')
### flow: $graph->get_graph_attribute('flow') // 'south'
$easy->set_attribute('flow',
$graph->get_graph_attribute('flow') // 'south');
if (defined(my $name = $graph->get_graph_attribute('name'))) {
$easy->set_attribute('label',$name);
}
Graph_Easy_view($easy, %options);
# print "Graph: ", $graph->is_directed ? "directed\n" : "undirected\n";
# print "Easy: ", $easy->is_directed ? "directed\n" : "undirected\n";
}
sub Graph_vertex_parent {
my ($graph, $v, %options) = @_;
my $cmp = $options{'cmp'} || cmp_func($graph->vertices);
my @parents = grep {$cmp->($_,$v)<=0} $graph->neighbours($v);
return $parents[0];
}
sub Graph_vertex_depth {
my ($graph, $v) = @_;
my $depth = 0;
while ($v = Graph_vertex_parent($graph,$v)) {
$depth++;
}
return $depth;
}
sub Graph_tree_root {
my ($graph) = @_;
if (defined (my $root = $graph->get_graph_attribute('root'))) {
return $root;
}
if (defined(my $root = $graph->get_graph_attribute('root_vertex'))) {
return $root;
}
foreach my $v ($graph->vertices) {
if ($v =~ /^0+$/) {
return $v; # 0 or 00 or 000 etc
}
}
if ($graph->has_vertex(1)) {
return 1;
}
croak "No tree root found";
}
# Note: This depends on the Graph_vertex_children() vertex numbering.
sub Graph_tree_height {
my ($graph, $root) = @_;
$root //= Graph_tree_root($graph);
### $root
my $height = 0;
my @pending = ($root);
my $depth = 0;
for (;;) {
@pending = map {Graph_vertex_children($graph,$_)} @pending;
last unless @pending;
$depth++;
}
return $depth;
}
# Return a list of arrayrefs [$v,$v,...] which are the vertices at
# successive depths of tree $graph. The first arrayref contains the tree root.
# Within each row vertices are sorted first by parent, then by given cmp.
sub Graph_vertices_by_depth {
my ($graph, %options) = @_;
my $root = $options{'root'} // Graph_tree_root($graph);
my @ret = ([$root]);
my $cmp = $options{'cmp'} || cmp_func($graph->vertices);
my %seen = ($root => 1);
for (;;) {
my @row = map { sort $cmp
grep {!$seen{$_}++}
$graph->neighbours($_) } @{$ret[-1]};
@row || last;
push @ret, \@row;
}
return @ret;
}
# return a list of all the descendents of vertex $v
# children are vertices compared numerically $child >= $v
sub Graph_tree_descendents {
my ($graph, $v) = @_;
my @ret;
my @pending = ($v);
while (@pending) {
@pending = map {Graph_vertex_children($graph,$_)} @pending;
push @ret, @pending;
}
return @ret;
}
sub cmp_numeric ($$) {
my ($a, $b) = @_;
return $a <=> $b;
}
sub cmp_alphabetic ($$) {
my ($a, $b) = @_;
return $a cmp $b;
}
sub cmp_func {
return (all_looks_like_number(@_) ? \&cmp_numeric : \&cmp_alphabetic);
}
sub Graph_tree_print {
my ($graph, %options) = @_;
### Graph_tree_print() ...
my $flow = ($options{'flow'} // 'down');
my $hat = '^';
my $slash = '/';
my $backslash = '\\';
if ($flow eq 'up') {
$hat = 'v';
$slash = '\\';
$backslash = '/';
}
{
# by successive adjustment
my $gap = 2;
my $sibling_gap = 1;
my $cmp = $options{'cmp'} || cmp_func($graph->vertices);
### numeric: $cmp==\&cmp_numeric
my @vertices = $graph->vertices;
my @vertices_by_depth = Graph_vertices_by_depth($graph, cmp => $cmp);
### @vertices_by_depth
my %column;
my %children;
foreach my $v (@vertices) {
$children{$v} = [ sort $cmp Graph_vertex_children($graph,$v,cmp=>$cmp) ];
$column{$v} = 0;
### children: "$v children ".join(',',@{$children{$v}})
}
my $are_siblings = sub {
my ($v1,$v2) = @_;
my $p1 = Graph_vertex_parent($graph,$v1, cmp=>$cmp) // return 0;
my $p2 = Graph_vertex_parent($graph,$v2, cmp=>$cmp) // return 0;
return $p1 eq $p2;
};
OUTER: for (my $limit = 0; $limit < 50; $limit++) {
# avoid overlaps within row
foreach my $depth (0 .. $#vertices_by_depth) {
my $aref = $vertices_by_depth[$depth];
foreach my $i (1 .. $#$aref) {
my $v1 = $aref->[$i-1];
my $v2 = $aref->[$i];
my $c = $column{$v1} + length($v1)
+ ($are_siblings->($v1,$v2) ? $sibling_gap : $gap);
if ($column{$v2} < $c) {
### overlap in row: "depth=$depth $v2 to column $c"
$column{$v2} = $c;
next OUTER;
}
}
}
# parent half-way along children,
# or children moved up if parent further along
foreach my $p (@vertices) {
my $children_aref = $children{$p};
next unless @$children_aref;
my $min = $column{$children_aref->[0]};
my $max = $column{$children_aref->[-1]} + length($children_aref->[-1]);
my $c = int(($max + $min - length($p))/2);
if ($column{$p} < $c) {
### parent: "parent p=$p move up to middle $c ($min to $max)"
$column{$p} = $c;
next OUTER;
}
if ($column{$p} > $c) {
my $v = $children_aref->[0];
$column{$v}++;
### parent: "parent $p at $column{$v} > mid $c, advance first child c=$v (".join(',',@$children_aref).") to column $column{$v}"
next OUTER;
}
}
# leading leaf child moves up to its sibling
foreach my $parent (@vertices) {
my $children_aref = $children{$parent};
my $v1 = $children_aref->[0] // next;
my $v2 = $children_aref->[1] // next;
next if @{$children{$v1}}; # want $v1 leaf
my $c = $column{$v2} - length($v1) - $sibling_gap;
if ($column{$v1} < $c) {
$column{$v1} = $c;
next OUTER;
}
}
last;
}
my $total_column = max(map {$column{$_}+length($_)} @vertices) + 3;
my @lines;
foreach my $depth (0 .. $#vertices_by_depth) {
my $aref = $vertices_by_depth[$depth];
my $c = 0;
my $line = '';
foreach my $v (@$aref) {
$column{$v} ||= 0;
while ($c < ($column{$v}||0)) {
$line .= " ";
$c++;
}
$line .= $v . " ";
$c += length($v)+1;
}
while ($c < $total_column) {
$line .= " ";
$c++;
}
my $count = @$aref;
$line .= "count $count\n";
push @lines, $line;
$line = '';
$c = 0;
if ($depth < $#vertices_by_depth) {
my @lines;
foreach my $v (@$aref) {
my $children_aref = $children{$v};
next unless @$children_aref;
my $min = $column{$children_aref->[0]};
my $max = $column{$children_aref->[-1]} + length($children_aref->[-1]);
if (@$children_aref > 1) {
if (length($children_aref->[0]) > 1) { $min++; }
if (length($children_aref->[-1]) > 1) { $max--; }
}
my $mid = int($column{$v} + length($v)/2);
while ($c < $min) { $line .= ' '; $c++; }
# while ($c < $max) { print($c == $mid ? '|' : '_'); $c++; }
while ($c < $max) {
$line .= ($c == $mid && @$children_aref != 2 ? '|'
: @$children_aref == 1 ? ' '
: $c == $min ? $slash
: $c == $max-1 ? $backslash
: $c == $mid ? ($max-$min <=3 && $flow eq 'up' ? ' ' : $hat)
: '-'); $c++;
}
}
}
$line .= "\n";
push @lines, $line;
}
if ($flow eq 'up') {
@lines = reverse @lines;
}
print @lines;
return;
}
{
my @vertices_by_depth = Graph_vertices_by_depth($graph,
cmp => $options{'cmp'});
### @vertices_by_depth
my @column;
foreach my $aref (reverse @vertices_by_depth) {
my $c = 0;
foreach my $v (@$aref) {
my @children = Graph_vertex_children($graph,$v);
if (@children) {
### vertex: "$v children @children"
$c = max($c,
ceil( sum(map {$column[$_] + (length($_)+1)/2} @children) / scalar(@children)
- (length($v)+1)/2 ));
}
$column[$v] = $c;
$c += length($v) + 1;
$c = max($c, map {$column[$_] + length($_)+1} Graph_tree_descendents($graph, $v));
}
}
my $total_column = max(map {($column[$_]||0)+length($_)} 0 .. $#column) + 3;
foreach my $aref (@vertices_by_depth) {
my $c = 0;
### columns: map {$column[$_]} @$aref
foreach my $v (@$aref) {
while ($c < $column[$v]) {
print " ";
$c++;
}
print $v," ";
$c += length($v)+1;
}
while ($c < $total_column) {
print " ";
$c++;
}
my $count = @$aref;
print "count $count\n";
}
return;
}
}
# use Smart::Comments;
sub Graph_tree_layout {
my ($graph, %options) = @_;
### Graph_tree_layout ...
my $v = $options{'v'} // Graph_tree_root($graph);
my $x = $options{'x'} || 0;
my $y = $options{'y'} || 0;
my $order = $options{'order'} || '';
my $align = $options{'align'} || '';
my $filled = $options{'filled'} // [];
my @children = Graph_vertex_children($graph,$v);
my @heights = map {Graph_tree_height($graph,$_)} @children;
my @order;
if ($order eq 'name') {
@order = sort {$graph->get_vertex_attribute($children[$a],'name')
cmp $graph->get_vertex_attribute($children[$b],'name')}
0 .. $#children;
} else {
@order = sort {$heights[$b] <=> $heights[$a]} 0 .. $#children;
}
my $h = (@children ? $heights[$order[0]]+1 : 0);
### $h
Y: for (;;) {
foreach my $i (0 .. $h) {
if ($filled->[$x+$i]->[$y]) {
$filled->[$x]->[$y] = 1;
$y++;
next Y;
}
}
last;
}
### place: "$v at $x,$y"
Graph_set_xy_points($graph, $v => [$x,-$y]);
$filled->[$x]->[$y] = 1;
foreach my $i (@order) {
Graph_tree_layout($graph, v=>$children[$i],
x=>$x+1, y=>$y++,
filled => $filled,
order => $order,
align => $align);
}
}
# no Smart::Comments;
#------------------------------------------------------------------------------
# vertices are coordinate strings "$x,$y" and edges along a square grid
# print an ascii form of the graph
#
sub Graph_xy_print {
my ($graph) = @_;
my @vertices = $graph->vertices;
my @points = map {[split /,/]} @vertices;
my @x = map {$_->[0]} @points;
my @y = map {$_->[1]} @points;
my $x_min = (@x ? min(@x) - 1 : 0);
my $x_max = (@x ? max(@x) + 1 : 0);
my $y_min = (@y ? min(@y) - 1 : 0);
my $y_max = (@y ? max(@y) + 1 : 0);
foreach my $y (reverse $y_min .. $y_max) {
printf "%3s ", '';
foreach my $x ($x_min .. $x_max) {
my $from = "$x,$y";
# vertical edge to above
print $graph->has_edge($from, $x.",".($y+1)) ? "| " : " ";
}
print "\n";
printf "%3d ", $y;
foreach my $x ($x_min .. $x_max) {
my $from = "$x,$y";
# horizontal edge to next
print $graph->has_vertex($from) ? "*" : " ";
print $graph->has_edge($from, ($x+1).",".$y) ? "---" : " ";
}
print "\n";
}
print " ";
foreach my $x ($x_min .. $x_max) {
printf "%4d", $x;
}
print "\n";
}
sub Graph_xy_print_triangular {
my ($graph) = @_;
my @vertices = $graph->vertices;
my @points = map {[split /,/]} @vertices;
my @x = map {$_->[0]} @points;
my @y = map {$_->[1]} @points;
my $x_min = (@x ? min(@x) - 1 : 0);
my $x_max = (@x ? max(@x) + 1 : 0);
my $y_min = (@y ? min(@y) - 1 : 0);
my $y_max = (@y ? max(@y) + 1 : 0);
foreach my $y (reverse $y_min .. $y_max) {
printf "%3s ", '';
foreach my $x ($x_min .. $x_max) {
my $from = "$x,$y";
# vertical edge to above
print $graph->has_edge($from, ($x ).",".($y+1)) ? "|" : " ";
print $graph->has_edge(($x).",".($y+1), ($x+1).",".($y)) ? "\\"
: $graph->has_edge($from, ($x+1).",".($y+1)) ? "/" : " ";
}
print "\n";
printf "%3d ", $y;
foreach my $x ($x_min .. $x_max) {
my $from = "$x,$y";
# horizontal edge to next
print $graph->has_vertex($from) ? "*"
: $graph->has_edge(($x-1).",".$y, ($x+1).",".$y) ? "-"
: " ";
print $graph->has_edge(($x-1).",".$y, ($x+1).",".$y)
|| $graph->has_edge($from, ($x+1).",".$y)
|| $graph->has_edge($from, ($x+2).",".$y) ? "-" : " ";
}
print "\n";
}
print " ", ($x_min&1 ? ' ' : '');
for (my $x = $x_min+($x_min&1); $x <= $x_max; $x+=2) {
printf "%4d", $x;
}
print "\n";
}
#------------------------------------------------------------------------------
our $HOG_directory = File::Spec->catdir(File::HomeDir->my_home, 'HOG');
# $coderef = make_tree_iterator_edge_aref()
# Return a function which iterates through trees in the form of edge arrayrefs.
# Each call to the function is
# $edge_aref = $coderef->();
# returning an arrayref [ [1,2], [2,3], ... ] of a tree, or undef at end of
# iteration.
#
# Optional key/value parameters are
# num_vertices_min => $integer \ min and max vertices in the trees
# num_vertices_max => $integer /
# degree_list => arrayref [ 1, 2, 4 ]
# degree_max => $integer
# degree_predicate => $coderef
#
sub make_tree_iterator_edge_aref {
my %option = @_;
require Graph::Graph6;
my $degree_predicate_aref;
if (defined (my $degree_list = $option{'degree_list'})) {
my @degree_predicate_array;
foreach my $degree (@$degree_list) {
$degree_predicate_array[$degree] = 1;
}
$degree_predicate_aref = \@degree_predicate_array;
} elsif (defined (my $degree_max = $option{'degree_max'})) {
my @degree_predicate_array;
foreach my $degree (1 .. $degree_max) {
$degree_predicate_array[$degree] = 1;
}
$degree_predicate_aref = \@degree_predicate_array;
}
my $num_vertices= ($option{'num_vertices'}
// $option{'num_vertices_min'}
// 1);
$num_vertices = max(1, $num_vertices); # no trees of 0 vertices
my $num_vertices_max = ($option{'num_vertices_max'}
// $num_vertices);
$num_vertices--;
my $fh;
return sub {
for (;;) {
if (! $fh) {
if (defined $num_vertices_max && $num_vertices >= $num_vertices_max) {
return;
}
$num_vertices++;
### open: $num_vertices
my $filename = File::Spec->catfile($HOG_directory,
sprintf('trees%02d.g6',
$num_vertices));
open $fh, '<', $filename
or die "Cannot open $filename: $!";
}
my @edges;
unless (Graph::Graph6::read_graph(fh => $fh,
num_vertices_ref => \my $file_num_vertices,
edge_aref => \@edges)) {
### EOF ...
close $fh or die;
undef $fh;
next;
}
my $edge_aref = \@edges;
if ($degree_predicate_aref
&& ! edge_aref_degree_check($edge_aref, $degree_predicate_aref)) {
### skip for degree_max ...
next;
}
return $edge_aref;
}
};
}
# Return an iterator $itfunc to be called as
# $edge_aref = $itfunc->()
# which iterates through all connected graphs.
# Parameters:
# num_vertices => integer
# num_vertices_min => integer
# num_vertices_max => integer
# num_edges_min => integer
# num_edges_max => integer
# connected => bool, default true
#
sub make_graph_iterator_edge_aref {
my %option = @_;
require Graph::Graph6;
my $num_vertices = ($option{'num_vertices'}
// $option{'num_vertices_min'}
// 1);
my $num_vertices_max = ($option{'num_vertices_max'}
// $option{'num_vertices'});
my $num_edges_min = $option{'num_edges_min'};
my $num_edges_max = $option{'num_edges_max'};
my @geng_edges_option;
if ($option{'verbose'}) {
push @geng_edges_option, '-v';
} else {
push @geng_edges_option, '-q';
}
if ($option{'connected'} // 1) {
push @geng_edges_option, '-c';
}
if (defined $num_edges_max || defined $num_edges_min) {
if (! defined $num_edges_min) { $num_edges_min = 0; }
if (! defined $num_edges_max) { $num_edges_max = '#'; }
push @geng_edges_option, "$num_edges_min:$num_edges_max";
}
### @geng_edges_option
$num_vertices--;
require IPC::Run;
my $fh;
return sub {
for (;;) {
if (! $fh) {
if (defined $num_vertices_max && $num_vertices >= $num_vertices_max) {
return;
}
$num_vertices++;
### open: $num_vertices
IPC::Run::start(['nauty-geng',
# '-l', # canonical
$num_vertices,
@geng_edges_option],
'<', File::Spec->devnull,
'|', ['sort'],
'>pipe', \*OUT);
$fh = \*OUT;
}
my @edges;
unless (Graph::Graph6::read_graph(fh => $fh,
num_vertices_ref => \my $file_num_vertices,
edge_aref => \@edges)) {
### EOF ...
close $fh or die;
undef $fh;
next;
}
my $edge_aref = \@edges;
return $edge_aref;
}
};
}
# Return true if the degrees of the nodes in $edge_aref all have
# arrayref $degree_predicate_aref->[$degree] true.
#
sub edge_aref_degree_check {
my ($edge_aref, $degree_predicate_aref) = @_;
my @vertex_degree;
foreach my $edge (@$edge_aref) {
my ($from, $to) = @$edge;
$vertex_degree[$from]++;
$vertex_degree[$to]++;
}
foreach my $degree (@vertex_degree) {
if (! $degree_predicate_aref->[$degree]) {
return 0;
}
}
return 1;
}
# $edge_aref is an arrayref [ [from,to], [from,to], ... ]
# where each vertex is integer 0 upwards
# Return a list (degree, degree, ...) of degree of each vertex
sub edge_aref_degrees {
my ($edge_aref) = @_;
### edge_aref_degrees: $edge_aref
my @vertex_degree;
foreach my $edge (@$edge_aref) {
my ($from, $to) = @$edge;
$vertex_degree[$from]++;
$vertex_degree[$to]++;
}
return map {$_//0} @vertex_degree;
}
# $edge_aref is an arrayref [ [from,to], [from,to], ... ]
# where each vertex is integer 0 upwards
# Return a list (degree, degree, ...) of distinct vertex degrees which occur
# in the graph.
sub edge_aref_degrees_distinct {
my ($edge_aref) = @_;
my @vertex_degree = edge_aref_degrees($edge_aref);
my %seen;
@vertex_degree = grep {! $seen{$_}++} @vertex_degree;
return sort {$a<=>$b} @vertex_degree;
}
# Trees by search.
# {
# my @parent = (undef, -1);
# my $v = 1;
# for (;;) {
# my $p = ++$parent[$v];
# ### at: "$v consider new parent $p"
# if ($p >= $v) {
# ### backtrack ...
# $v--;
# if ($v < 1) { last; }
# $p = $parent[$v]; # unparent this preceding v
# $num_children[$p]--;
# next;
# }
#
# if ($num_children[$p] >= ($p==0 ? 4 : 3)) {
# next;
# }
#
# $num_vertices = $v;
# $process_tree->();
#
# if ($v < $num_vertices_limit) {
# # descend
# $num_children[$p]++;
# # $num_children[$p] == grep {$_==$p} @parent[1..$v] or die;
# $num_children[$v] = 0;
# $v++;
# $parent[$v] = -1;
# $num_vertices = $v;
# }
# }
# }
# Tree iterator by parent.
# {
# @parent = (undef);
# @num_children = (0);
# my $v = 0;
# for (;;) {
# $num_children[$v]++;
# my $new_v = $v + $num_children[$v];
# ### at: "$v consider new children $num_children[$v]"
#
# if ($num_children[$v] > ($v==0 ? 4 : 3)
# || $new_v > $num_vertices_limit) {
# ### backtrack ...
# $v = $parent[$v] // last;
# next;
# }
#
# # add children
# foreach my $i (1 .. $num_children[$v]) {
# my $c = $v + $i;
# $parent[$c] = $v;
# $num_children[$c] = 0;
# }
# $v = $new_v-1;
# $num_vertices = $v;
# $process_tree->();
# }
# }
#------------------------------------------------------------------------------
use constant::defer hog_directory => sub {
require File::Spec;
require File::HomeDir;
File::Spec->catdir(File::HomeDir->my_home, 'HOG');
};
use constant::defer hog_all_filename => sub {
require File::Spec;
File::Spec->catdir(hog_directory, 'all.g6');
};
use constant::defer hog_mmap_ref => sub {
require File::Map;
my $mmap;
File::Map::map_file ($mmap, hog_all_filename());
return \$mmap;
};
# $str is a string of graph6 in canonical labelling.
# Return true if it is in the House of Graphs, based on grepping the all.g6
# file (hog_all_filename()).
sub hog_grep {
my ($str) = @_;
require File::Slurp;
### hog_grep(): $str
$str =~ s/\n$//;
my $mmap_ref = hog_mmap_ref();
if ($$mmap_ref =~ /^\Q$str\E$/m) {
$str =~ s/\n+$//g;
foreach my $filename (glob(File::Spec->catfile(hog_directory(), 'graph_*.g6'))) {
if (defined (my $file_str = File::Slurp::read_file($filename, err_mode=>'quiet'))) {
$file_str =~ s/\n+$//g;
if ($file_str eq $str) {
if ($filename =~ m{graph_([^/]*)\.g6$}) {
return $1;
} else {
return $filename;
}
}
}
}
return -1;
}
}
# $num is a House of Graphs graph ID number.
# Return the local filename for its graph6.
# There's no check whether the file actually exists.
sub hog_num_to_filename {
my ($num) = @_;
require File::Spec;
File::Spec->catfile(hog_directory(), "graph_$num.g6");
}
sub hog_compare {
my ($id, $g6_str) = @_;
require File::Slurp;
my $filename = hog_num_to_filename($id);
my $file_str = File::Slurp::read_file($filename);
my $canon_g6_str = graph6_str_to_canonical($g6_str);
my $canon_file_str = graph6_str_to_canonical($file_str);
if ($g6_str ne $file_str) {
print "id=$id wrong\n";
print "string $g6_str";
print "file $file_str";
print "canon string $canon_g6_str";
print "canon file $canon_file_str";
croak "wrong";
}
}
sub hog_id_to_url {
my ($id) = @_;
# ENHANCE-ME: maybe escape against some bad id string
return "https://hog.grinvin.org/ViewGraphInfo.action?id=$id";
}
# hog_searches_html($graph,$graph,...)
# Create a /tmp/USERNAME/hog-searches.html of forms to search hog for each
# $graph. Each $graph can be either Graph.pm or Graph::Easy.
#
# The hog-searches.html is a bit rough, and requires you select the 0.g6,
# 1.g6, etc file to search for. The HOG server expects a file upload, and
# don't think can induce a browser to do a file-like POST other than by
# selecting a file. Some Perl code POST could do it easily, but the idea is
# to present a range of searches and you might only do a few.
#
sub hog_searches_html {
my @graphs = @_;
### hog_searches_html() ...
require HTML::Entities;
require File::Spec;
require File::Temp;
require POSIX;
my $dir = File::Spec->catdir('/tmp', POSIX::cuserid());
mkdir $dir;
my $html_filename = File::Spec->catfile($dir, 'hog-searches.html');
my $hog_url = 'https://hog.grinvin.org';
# $hog_url = 'http://localhost:10000'; # for testing
my @names;
open my $h, '>', $html_filename or die;
print $h <<'HERE';
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<body>
HERE
my %seen_canonical;
foreach my $i (0 .. $#graphs) {
my $graph = $graphs[$i];
### graph: "$graph"
if (! ref $graph) {
### convert graph6 string ...
$graph = Graph_from_graph6_str($graph);
} elsif (! blessed($graph)) {
### convert edge_aref ...
if (ref $graph) {
$graph = edge_aref_to_Graph_Easy($graph);
} else {
$graph = Graph_from_graph6_str($graph);
}
}
my $png_fh = File::Temp->new;
my $png_filename = $png_fh->filename;
my $graph6_str;
if ($graph->isa('Graph::Easy')) {
require Graph::Easy::As_graph6;
$graph6_str = $graph->as_graph6;
} else {
$graph6_str = Graph_to_graph6_str($graph);
}
my $graph6_size = length $graph6_str;
my $num_vertices = $graph->vertices;
my $num_edges = $graph->edges;
my $name;
my $flow = 'south';
my $vertex_name_type;
if ($graph->isa('Graph::Easy')) {
$name = $graph->get_attribute('label');
# FIXME: custom attributes?
# $vertex_name_type = $graph->get_attribute('graph','vertex_name_type');
} else {
$name = $graph->get_graph_attribute('name');
$vertex_name_type = $graph->get_graph_attribute('vertex_name_type');
$flow = $graph->get_graph_attribute('flow') // $flow;
}
$vertex_name_type //= '';
$name //= '';
$names[$i] = $name;
my $graph6_canonical = graph6_str_to_canonical($graph6_str);
my $canonical = $graph6_canonical;
if (length($canonical) > 30) {
$canonical = '';
} else {
$canonical = "<br> canonical "
. HTML::Entities::encode_entities($canonical);
}
if (defined(my $prev = $seen_canonical{$graph6_canonical})) {
print "g$i $name\n REPEAT g$prev $names[$prev]\n";
print $h "<br> repeat of $seen_canonical{$graph6_canonical} ",
HTML::Entities::encode_entities($names[$prev]),
"\n";
} else {
$seen_canonical{$graph6_canonical} = $i;
}
my $got = '';
if (my $num = hog_grep($graph6_canonical)) {
my $str = $graph6_canonical;
$str =~ s/\n+$//;
print "g$i HOG got $str n=$num_vertices",
($num eq '-1' ? '' : " id=$num"),
" $name\n";
if ($num eq '-1') {
my $filename = HTML::Entities::encode_entities(hog_all_filename());
$got = "<br> got in $filename\n";
} else {
my $url = hog_id_to_url($num);
$num = HTML::Entities::encode_entities($num);
$got = "<br> got <a href=\"$url\">HOG id $num</a>\n";
}
}
print $h <<"HERE";
<hr width="100%">
<p>
$graph6_size bytes,
$num_vertices vertices,
$num_edges edges
$name$canonical$got
HERE
if ($num_vertices == 0) {
print $h "empty\n";
}
print $h <<"HERE";
<FORM name="DoSearchGraphFromGraph6String"
action="$hog_url/DoSearchGraphFromGraph6String.action"
method="post">
<INPUT type="text" name="graph6String" value="$graph6_str"/>
<INPUT type="submit"
value="Search"/>
</FORM>
HERE
if ($num_vertices <= 60) {
my $is_xy = $graph->isa('Graph')
&& ($graph->get_graph_attribute('vertex_name_type_xy')
|| $graph->get_graph_attribute('xy')
|| do {
my $type = $graph->get_graph_attribute('vertex_name_type');
defined $type && $type =~ /^xy/ }
|| do {
my ($v) = sort $graph->vertices;
defined $v && defined($graph->get_vertex_attribute($v,'x')) });
if ($is_xy || 1) {
### write with graphviz2 neato ...
my $graphviz2 = Graph_to_GraphViz2($graphs[$i]);
$graphviz2->run(format => 'png',
output_file=>$png_filename,
driver => 'neato');
### dot_input: $graphviz2->dot_input
} elsif (1) {
### write with graphviz2 dot ...
my $graphviz2 = Graph_to_GraphViz2($graphs[$i]);
$graphviz2->run(format => 'png',
output_file=>$png_filename);
} else {
my $easy = $graph;
if ($graph->isa('Graph')) {
### Graph num nodes ...
my $graph = $graph->copy;
foreach my $v ($graph->vertices) {
$graph->delete_vertex_attribute($v,'xy');
}
require Graph::Convert;
$easy = Graph::Convert->as_graph_easy($graph);
}
# Graph_Easy_blank_labels($easy);
foreach my $v (1,0) {
if (defined($easy->node($v))) {
$easy->set_attribute('root',$v); # for as_graphviz()
$easy->set_attribute('flow',$flow); # for as_graphviz()
}
}
### Graph-Easy num nodes: scalar($easy->nodes)
$easy->set_attribute('x-dot-start','1');
my $graphviz = $easy->as_graphviz;
# $graphviz =~ s/node \[/node [\n height=.08,\n width=.08,\n fixedsize=1,/;
# print $graphviz;
require IPC::Run;
IPC::Run::run(['dot','-Tpng',
],
'<',\$graphviz, '>',$png_filename);
# IPC::Run::run(['neato','-Tpng'], '<',\$graphviz, '>',$png_filename);
# IPC::Run::run(['fdp','-Tpng'], '<',\$graphviz, '>',$png_filename);
# print $easy->as_ascii;
}
require File::Slurp;
my $png = File::Slurp::read_file($png_filename);
require URI::data;
my $png_uri = URI->new("data:");
$png_uri->data($png);
$png_uri->media_type('image/png');
# my = URI::data->new($png,'image/png');
print $h qq{<img src="$png_uri">\n};
}
}
print $h <<'HERE';
</body>
</html>
HERE
close $h or die;
print scalar(@graphs)," graphs\n";
print "iceweasel file://$html_filename >/dev/null 2>&1 &\n";
}
# blank out all labels of a Graph::Easy
sub Graph_Easy_blank_labels {
my ($easy) = @_;
foreach my $node ($easy->nodes) {
$node->set_attribute(label => ' ');
}
}
sub edge_aref_to_Graph_Easy {
my ($edge_aref) = @_;
### $edge_aref
require Graph::Easy;
my $easy = Graph::Easy->new (undirected => 1);
foreach my $edge (@$edge_aref) {
scalar(@$edge) == 2 or die "bad edge_aref";
my ($from, $to) = @$edge;
($from =~ /^[0-9]+$/ && $to =~ /^[0-9]+$/) or die "bad edge_aref";
$easy->add_edge($from,$to);
}
return $easy;
}
sub edge_aref_to_Graph {
my ($edge_aref) = @_;
require Graph;
my $graph = Graph->new (undirected => 1);
$graph->add_edges(@$edge_aref);
return $graph;
}
sub edge_aref_string {
my ($edge_aref) = @_;
return join(',', map{join('-',@$_)} @$edge_aref)
. ' ['.scalar(@$edge_aref).' edges]';
}
# Create a file /tmp/USERNAME/hog-upload.html which is an upload of $graph.
# This uses the HOG add-a-graph by drawing. Log-in first, then click Upload
# in hog-upload.html.
#
# The upload is an adjacency matrix and vertex locations. These are the
# text fields in the HTML, but are likely to be too big to see anything
# useful.
# Vertex locations are from Graph_vertex_xy($graph, ...).
# The server draws straight-line edges between locations.
# hog-upload.html includes a simple png image so you can preview how it
# ought to come out. The Upload click goes to the usual HOG page to enter a
# name and comment. You don't see the image in HOG until after that, but if
# it goes badly wrong you can always delete the graph.
#
sub hog_upload_html {
my ($graph, %options) = @_;
require POSIX;
require File::Spec;
require File::Temp;
my $dir = File::Spec->catdir('/tmp', POSIX::cuserid());
mkdir $dir;
my $html_filename = File::Spec->catfile($dir, 'hog-upload.html');
# my $png_filename = File::Spec->catfile($dir, 'hog-upload.png');
my $png_fh = File::Temp->new;
my $png_filename = $png_fh->filename;
my $hog_url = 'https://hog.grinvin.org';
# $hog_url = 'http://localhost'; # for testing
my @vertices = MyGraphs::Graph_sorted_vertices($graph);
my $name = $graph->get_graph_attribute('name') // '';
my $num_vertices = scalar(@vertices);
my $num_edges = $graph->edges;
print "graph $name\n";
print "$num_vertices vertices, $num_edges edges\n";
my $yscale = $options{'yscale'} || 1;
if ($graph->get_graph_attribute('is_xy_triangular')) {
$yscale *= sqrt(3);
}
my @points = map { my ($x,$y) = MyGraphs::Graph_vertex_xy($graph,$_)
or croak("no X,Y coordinates for vertex ",$_);
[$x,$yscale*$y]
} @vertices;
### @points
if (my $a = $options{'rotate_degrees'}) {
$a = Math::Trig::deg2rad($a);
my $s = sin($a);
my $c = cos($a);
@points = map {
[ $_->[0] * $c - $_->[1] * $s,
$_->[0] * $s + $_->[1] * $c ]
} @points;
}
my @x = map {$_->[0]} @points;
my @y = map {$_->[1]} @points;
my $size = max( max(@x)-min(@x), max(@y)-min(@y) );
require Geometry::AffineTransform;
my $affine = Geometry::AffineTransform->new;
$affine->translate( -(max(@x)+min(@x))/2, -(max(@y)+min(@y))/2 );
$affine->scale(1/$size, -1/$size); # Y down the page
$affine->scale(380, 380);
$affine->translate(200, 200);
@points = map {[$affine->transform(@$_)]} @points;
@points = map {[map {POSIX::round($_)} @$_]} @points;
@x = map {$_->[0]} @points;
@y = map {$_->[1]} @points;
print "transformed x ",min(@x)," to ",max(@x),
" y ",min(@y)," to ",max(@y),"\n";
require Image::Base::GD;
my $image = Image::Base::GD->new (-width => 400, -height => 400);
$image->rectangle(0,0, 400,400, 'white', 1);
$image->rectangle(0,0, 399,399, 'blue');
foreach my $from (0 .. $#vertices) {
foreach my $to (0 .. $#vertices) {
if ($graph->has_edge($vertices[$from], $vertices[$to])) {
$image->line(@{$points[$from]}, @{$points[$to]}, 'red');
}
}
}
foreach my $from (0 .. $#vertices) {
my ($x,$y) = @{$points[$from]};
$image->ellipse($x-1,$y-1, $x+1,$y+1, 'black');
}
$image->save($png_filename);
require File::Slurp;
my $png = File::Slurp::read_file($png_filename);
require URI::data;
my $png_uri = URI->new("data:");
$png_uri->data($png);
$png_uri->media_type('image/png');
# my = URI::data->new($png,'image/png');
# stringize the points
@points = map {join('-',@$_).';'} @points;
### @points
unless (list_is_all_distinct_eq(@points)) {
die "oops, some point coordinates have rounded together";
}
my $coordinateString = join('',@points);
### $coordinateString
# 0100000000000000%0D%0A
# 1010000000000000%0D%0A
# 0101000000000000%0D%0A
# 0010100000000000%0D%0A
# 0001010000000000%0D%0A
# 0000101010001000%0D%0A
# 0000010100000000%0D%0A
# 0000001010000000%0D%0A
# 0000010100000000%0D%0A
# 0000000000101000%0D%0A
# 0000000001010000%0D%0A
# 0000000000100001%0D%0A
# 0000010001000000%0D%0A
# 0000000000000010%0D%0A
# 0000000000000101%0D%0A
# 0000000000010010
my @adjacencies = map {
my $from = $_;
join('', map {$graph->has_edge($from,$_) ? 1 : 0} @vertices)
} @vertices;
### @adjacencies
my $adjacencyString = join("\r\n",@adjacencies);
require HTML::Entities;
my @names;
$name = HTML::Entities::encode_entities($name);
my $upsize = length($adjacencyString) + length($coordinateString) + 20;
print "upload size $upsize bytes\n";
open my $h, '>', $html_filename or die;
print $h <<"HERE";
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<body>
<h2>Upload</h2>
<p>
$name
<br>
$num_vertices vertices, $num_edges edges, size $upsize bytes
<br>
<form action="$hog_url/DoUploadGraph.action" method="POST">
<textarea name="adjacencyString" value="">$adjacencyString</textarea>
<br>
<input name="coordinateString" value="$coordinateString">
<br>
<input type=submit value="Upload">
</form>
<p>
<img width=400 height=400 src="$png_uri">
</p>
</body>
</html>
HERE
close $h or die;
print "iceweasel file://$html_filename >/dev/null 2>&1 &\n";
}
# Return true of all arguments are different, as compared by "eq".
sub list_is_all_distinct_eq {
my %seen;
foreach (@_) {
if ($seen{$_}++) {
return 0;
}
}
return 1;
}
#------------------------------------------------------------------------------
# nauty bits
sub graph6_view {
my ($g6_str, %options) = @_;
my $graph = Graph_from_graph6_str($g6_str);
my $name = $options{'name'};
if (! defined $name || $name eq '') {
my $num_vertices = $graph->vertices;
my $num_edges = $graph->edges;
$graph->set_graph_attribute
(name => "$num_vertices vertices, $num_edges edges");
}
Graph_view($graph);
}
sub graph6_str_to_canonical {
my ($g6_str, %options) = @_;
### graph6_str_to_canonical(): $g6_str
# num_vertices == 0 is already canonical and nauty-labelg doesn't like to
# crunch that
if ($g6_str =~ /^\?/) {
return $g6_str;
}
unless ($g6_str =~ /\n$/) { $g6_str .= "\n"; }
if ($g6_str =~ /\n.*\n/s) { croak "multiple newlines in g6 string"; }
my $canonical;
my $err;
require IPC::Run;
if (! IPC::Run::run
(['nauty-labelg',
(($options{'format'}||'') eq 'sparse6' ? '-s'
: '-g'), # graph6 output
# '-i2',
],
'<',\$g6_str,
'>',\$canonical,
'2>',\$err)) {
die "nauty-labelg error: ",$canonical,$err;
}
return $canonical;
}
sub Graph_to_sparse6_str {
my ($graph) = @_;
require Graph::Writer::Sparse6;
my $writer = Graph::Writer::Sparse6->new;
open my $fh, '>', \my $str or die;
$writer->write_graph($graph, $fh);
return $str;
}
sub Graph_to_graph6_str {
my ($graph, %options) = @_;
require Graph::Writer::Graph6;
my $writer = Graph::Writer::Graph6->new
(format => ($options{'format'}||'graph6'));
open my $fh, '>', \my $str or die;
$writer->write_graph($graph, $fh);
return $str;
}
# $str is a graph6 or sparse6 string
sub Graph_from_graph6_str {
my ($str) = @_;
require Graph::Reader::Graph6;
my $reader = Graph::Reader::Graph6->new;
open my $fh, '<', \$str or die;
return $reader->read_graph($fh);
}
# $filename is a file containing graph6 or sparse6
sub Graph_from_graph6_filename {
my ($filename) = @_;
require Graph::Reader::Graph6;
my $reader = Graph::Reader::Graph6->new;
open my $fh, '<', $filename or die 'Cannot open ',$filename,': ',$!;
return $reader->read_graph($fh);
}
# return true if Graph.pm graphs $g1 and $g2 are isomorphic
sub Graph_is_isomorphic {
my ($g1, $g2) = @_;
my $g1_str = graph6_str_to_canonical(Graph_to_graph6_str($g1));
my $g2_str = graph6_str_to_canonical(Graph_to_graph6_str($g2));
return $g1_str eq $g2_str;
}
sub Graph_from_edge_aref {
my ($edge_aref, %options) = @_;
my $num_vertices = delete $options{'num_vertices'};
my $graph = Graph->new (undirected => 1);
$graph->add_vertices (0 .. ($num_vertices||0)-1);
foreach my $edge (@$edge_aref) {
scalar(@$edge) == 2 or die "bad edge_aref";
my ($from, $to) = @$edge;
($from =~ /^[0-9]+$/ && $to =~ /^[0-9]+$/) or die "bad edge_aref";
$graph->add_edge($from,$to);
}
return $graph;
}
sub Graph_from_vpar {
my ($vpar, @options) = @_;
require Graph;
my $graph = Graph->new (@options);
$graph->add_vertices(1 .. $#$vpar);
foreach my $v (1 .. $#$vpar) {
if ($vpar->[$v]) {
$graph->add_edge ($v, $vpar->[$v]);
} else {
$graph->set_graph_attribute('root',$v);
}
}
if ($graph->is_directed) {
$graph->set_graph_attribute('flow','north');
}
return $graph;
}
sub Graph_to_vpar {
my ($graph, $root) = @_;
$root //= Graph_tree_root($graph);
### Graph_to_vpar() ...
### $root
my @vertices = sort $graph->vertices;
my @vpar = (undef, (0) x scalar(@vertices));
unshift @vertices, undef;
### @vertices
my %vertex_to_v;
foreach my $v (1 .. $#vertices) { $vertex_to_v{$vertices[$v]} = $v; }
my %seen;
$vpar[$vertex_to_v{$root}] = 0;
$seen{$root} = 1;
my @pending = ($root);
while (@pending) {
my $vertex = pop @pending;
my $v = $vertex_to_v{$vertex};
### vertex: "$vertex v=$v"
my @neighbours = $graph->neighbours($vertex);
### @neighbours
my $p = 0;
$seen{$vertex} = 1;
$vpar[$v] = 0;
foreach my $neighbour (@neighbours) {
if ($seen{$neighbour}) {
$p = $vertex_to_v{$neighbour};
$vpar[$v] = $p;
### set: "$v parent $p"
} else {
push @pending, $neighbour;
}
}
}
### @vpar
return \@vpar;
}
sub Graph_vpar_str {
my ($graph) = @_;
my $vpar = Graph_to_vpar($graph);
my $str = "[";
foreach my $v (1 .. $#$vpar) {
$str .= ($vpar->[$v] // 'undef');
if ($v != $#$vpar) { $str .= ","; }
}
$str .= "]";
}
sub Graph_print_vpar {
my ($graph) = @_;
my $vpar = Graph_to_vpar($graph);
print "[";
foreach my $v (1 .. $#$vpar) {
print $vpar->[$v] // 'undef';
if ($v != $#$vpar) { print ","; }
}
print "]\n";
}
# synchronous => 1, wait for viewer to exit before returning.
sub vpar_view {
my ($vpar, %options) = @_;
### Graph_view(): %options
my $graphviz2 = vpar_to_GraphViz2($vpar, %options);
GraphViz2_view ($graphviz2, %options);
}
sub vpar_name {
my ($vpar) = @_;
my $str = 'N='.$#$vpar.' vpar';
my $sep = ' ';
foreach my $i (1..$#$vpar) {
$str .= $sep;
if (length($str) >= 45) {
$str .= '...';
return $str;
}
$str .= $vpar->[$i];
$sep = ',';
}
return $str;
}
sub vpar_to_GraphViz2 {
my ($vpar, %options) = @_;
### vpar_to_GraphViz2(): %options
require GraphViz2;
my $name = $options{'name'} // vpar_name($vpar);
my $flow = ($options{'flow'} // 'up');
my $graphviz2 = GraphViz2->new
(global => { directed => 1 },
graph => { label => $name,
rankdir => ($flow eq 'down' ? 'TB'
: $flow eq 'up' ? 'BT'
: $flow),
ordering => 'out',
},
node => { margin => 0, # cf default 0.11,0.055
},
);
foreach my $v (1 .. $#$vpar) {
$graphviz2->add_node(name => $v,
margin => '0.04,0.03', # cf default 0.11,0.055
height => '0.1', # inches, minimum
width => '0.1', # inches, minimum
);
}
foreach my $from (1 .. $#$vpar) {
if (my $to = $vpar->[$from]) {
$graphviz2->add_edge(from => $from, to => $to);
}
}
# roots in cluster at same rank so aligned horizontally
$graphviz2->push_subgraph (subgraph => {rank => 'same'});
foreach my $v (1 .. $#$vpar) {
unless ($vpar->[$v]) {
$graphviz2->add_node(name => $v);
}
}
$graphviz2->pop_subgraph;
return $graphviz2;
}
#------------------------------------------------------------------------------
# triangles
# ($a,$b,$c) are vertices of a triangle in $graph.
# a
# / \
# b---c
# Return true if this is an even triangle. For an even triangle every other
# vertex in the graph has an edge going to an even number of the vertices
# a,b,c. This means either no edges to them, or edges to exactly 2 of them.
#
sub Graph_triangle_is_even {
my ($graph, $a,$b,$c) = @_;
### Graph_triangle_is_even(): "$a $b $c"
foreach my $v ($graph->vertices) {
next if $v eq $a || $v eq $b || $v eq $c;
my $count = (($graph->has_edge($v,$a) ? 1 : 0)
+ ($graph->has_edge($v,$b) ? 1 : 0)
+ ($graph->has_edge($v,$c) ? 1 : 0));
### count: "$v is $count"
unless ($count == 0 || $count == 2) {
### triangle odd ...
return 0;
}
}
### triangle even ...
return 1;
}
# $graph is a Graph.pm.
# Call $stop = $callback->($a,$b,$c) for each triangle in $graph.
# If the return $stop is true then stop the search.
# The return is the $stop value, or undef at end of search.
#
# c
# / \
# a---b
#
# Triangles are found one way only, so if a,b,c then no calls also for
# permutations like b,a,c. It's unspecified exactly which vertices are the
# $a,$b,$c in the callback (though the current code has then in ascending
# alphabetical order).
#
sub Graph_triangle_search {
my ($graph, $callback) = @_;
foreach my $a ($graph->vertices) {
my @a_neighbours = sort $graph->neighbours($a);
foreach my $bi (0 .. $#a_neighbours-2) {
my $b = $a_neighbours[$bi];
next if $b lt $a;
foreach my $ci ($bi+1 .. $#a_neighbours-1) {
my $c = $a_neighbours[$ci];
if ($graph->has_edge($b,$c)) {
if (my $stop = $callback->($a,$b,$c)) {
return $stop;
}
}
}
}
}
return undef;
}
# $graph is a Graph.pm.
# ($a,$b,$c) = Graph_find_triangle($graph);
# Return a list of vertices which are a triangle within $graph.
# If no triangles then return an empty list;
# c
# / \
# a---b
sub Graph_find_triangle {
my ($graph) = @_;
my @ret;
Graph_triangle_search($graph, sub {@ret = @_});
return @ret;
}
# $graph is a Graph.pm.
# Return true if $graph contains a triangle.
sub Graph_has_triangle {
my ($graph) = @_;
return Graph_triangle_search($graph, sub {1});
}
# $graph is a Graph.pm.
# Return the number of triangles in $graph.
sub Graph_triangle_count {
my ($graph) = @_;
my $count = 0;
Graph_triangle_search($graph, sub { $count++; return 0});
return $count;
}
#------------------------------------------------------------------------------
# Claws
# $graph is a Graph.pm.
# Call $stop = $callback->($a,$b,$c,$d) for each claw in $graph.
# $a is the centre.
# If the return $stop is true then stop the search.
# The return is the $stop value, or undef at end of search.
# b
# /
# a--c
# \
# d
sub Graph_claw_search {
my ($graph, $callback) = @_;
foreach my $a ($graph->vertices) {
my @a_neighbours = $graph->neighbours($a);
foreach my $bi (0 .. $#a_neighbours-2) {
my $b = $a_neighbours[$bi];
foreach my $ci ($bi+1 .. $#a_neighbours-1) {
my $c = $a_neighbours[$ci];
next if $graph->has_edge($b,$c);
foreach my $di ($ci+1 .. $#a_neighbours) {
my $d = $a_neighbours[$di];
next if $graph->has_edge($b,$d) || $graph->has_edge($c,$d);
if (my $stop = $callback->($a,$b,$c,$d)) {
return $stop;
}
}
}
}
}
return undef;
}
# $graph is a Graph.pm.
# ($a,$b,$c,$d) = Graph_find_claw($graph);
# Return a list of vertices which are a claw (a 4-star) within $graph, as an
# induced subgraph. $a is the centre.
# If no claws then return an empty list;
# b
# /
# a--c
# \
# d
sub Graph_find_claw {
my ($graph) = @_;
my @ret;
Graph_claw_search($graph, sub {@ret = @_});
return @ret;
}
# $graph is a Graph.pm.
# Return true if $graph contains a claw (star-4) as an induced subgraph.
sub Graph_has_claw {
my ($graph) = @_;
return Graph_claw_search($graph, sub {1});
}
# $graph is a Graph.pm.
# Return the number of induced claws in $graph.
sub Graph_claw_count {
my ($graph) = @_;
my $count = 0;
Graph_claw_search($graph, sub { $count++; return 0});
return $count;
}
#------------------------------------------------------------------------------
# Return a list of how many vertices at depths 0 etc down from $root.
# The first width is depth=0 which is $root itself so width=1.
sub Graph_width_list {
my ($graph, $root) = @_;
my @widths;
my %seen;
my @pending = ($root);
while (@pending) {
push @widths, scalar(@pending);
my @new_pending;
foreach my $v (@pending) {
$seen{$v} = 1;
my @children = $graph->neighbours($v);
@children = grep {! $seen{$_}} @children;
push @new_pending, @children;
}
@pending = @new_pending;
}
return @widths;
}
# $graph is a Graph.pm.
# Return true if all vertices of $graph have same degree.
#
sub Graph_is_regular {
my ($graph) = @_;
my $degree;
foreach my $v ($graph->vertices) {
my $d = $graph->degree($v);
$degree //= $d;
if ($d != $degree) { return 0; }
}
return 1;
}
# $graph and $subgraph are Graph.pm objects.
# Return true if $subgraph is a subgraph of $graph.
# This is a check of graph structure. The vertex names in the two can be
# different.
#
sub Graph_is_subgraph {
my ($graph, $subgraph) = @_;
my $num_vertices = $graph->vertices;
my $subgraph_num_vertices = $subgraph->vertices;
edge_aref_is_subgraph(edge_aref_from_Graph($graph),
edge_aref_from_Graph($subgraph),
num_vertices => $num_vertices,
subgraph_num_vertices => $subgraph_num_vertices);
}
sub edge_aref_from_Graph {
my ($graph) = @_;
### edge_aref_from_Graph(): "$graph"
my @vertices = sort $graph->vertices;
my %vertices = map { $vertices[$_] => $_ } 0 .. $#vertices;
my @edges = $graph->edges;
return [ map { my ($from,$to) = @$_;
[ $vertices{$from},$vertices{$to} ]
} @edges ];
}
sub Graph_is_induced_subgraph {
my ($graph, $subgraph, %options) = @_;
my @graph_vertices = sort $graph->vertices;
my @subgraph_vertices = sort $subgraph->vertices;
### @graph_vertices
### @subgraph_vertices
my @ret;
my @used = (0) x (scalar(@graph_vertices) + 1);
my @map = (-1) x (scalar(@subgraph_vertices) + 1);
my $pos = 0;
OUTER: for (;;) {
$used[$map[$pos]] = 0;
### undo use: "used=".join(',',@used)
for (;;) {
my $m = ++$map[$pos];
### $m
if ($m > $#graph_vertices) {
$pos--;
### backtrack to pos: $pos
if ($pos < 0) {
last OUTER;
}
next OUTER;
}
if (! $used[$m]) {
$used[$m] = 1;
last;
}
### used ...
}
### incremented: "pos=$pos map=".join(',',@map)." used=".join(',',@used)
if ($graph->vertex_degree($graph_vertices[$map[$pos]])
< $subgraph->vertex_degree($subgraph_vertices[$pos])) {
### graph degree smaller than subgraph ...
next;
}
foreach my $p (0 .. $pos-1) {
### consider: "pos=$pos p=$p graph $graph_vertices[$map[$p]] to $graph_vertices[$map[$pos]] subgraph $subgraph_vertices[$p] to $subgraph_vertices[$pos]"
my $gedge = !! $graph->has_edge ($graph_vertices[$map[$p]],
$graph_vertices[$map[$pos]]);
my $sedge = !! $subgraph->has_edge($subgraph_vertices[$p],
$subgraph_vertices[$pos]);
if ($gedge != $sedge) {
next OUTER;
}
}
# good for this next vertex at $pos, descend
if (++$pos > $#subgraph_vertices) {
# print "found:\n";
# foreach my $p (0 .. $#subgraph_vertices) {
# print " $subgraph_vertices[$p] <-> $graph_vertices[$map[$p]]\n";
# }
if ($options{'all_maps'}) {
push @ret, { map {$subgraph_vertices[$_] => $graph_vertices[$map[$_]]}
0 .. $#subgraph_vertices};
### new map: $ret[-1]
$pos--;
} else {
return join(', ',
map {"$subgraph_vertices[$_]=$graph_vertices[$map[$_]]"}
0 .. $#subgraph_vertices);
}
}
$map[$pos] = -1;
}
if ($options{'all_maps'}) {
return @ret;
} else {
return 0;
}
}
sub edge_aref_is_induced_subgraph {
my ($edge_aref, $subgraph_edge_aref) = @_;
if (@$edge_aref < @$subgraph_edge_aref) {
return 0;
}
my @degree;
my @neighbour;
foreach my $edge (@$edge_aref) {
$neighbour[$edge->[0]][$edge->[1]] = 1;
$neighbour[$edge->[1]][$edge->[0]] = 1;
$degree[$edge->[0]]++;
$degree[$edge->[1]]++;
}
### @degree
my @subgraph_degree;
my @subgraph_neighbour;
foreach my $edge (@$subgraph_edge_aref) {
$subgraph_neighbour[$edge->[0]][$edge->[1]] = 1;
$subgraph_neighbour[$edge->[1]][$edge->[0]] = 1;
$subgraph_degree[$edge->[0]]++;
$subgraph_degree[$edge->[1]]++;
}
### @subgraph_degree
{
my @degree_sorted = sort {$b<=>$a} @degree; # descending
my @subgraph_degree_sorted = sort {$b<=>$a} @subgraph_degree;
foreach my $i (0 .. $#subgraph_degree_sorted) {
if ($subgraph_degree_sorted[$i] > $degree_sorted[$i]) {
return 0;
}
}
}
my $num_vertices = scalar(@neighbour);
my $subgraph_num_vertices = scalar(@subgraph_neighbour);
my @used = (0) x ($num_vertices + 1);
my @map = (-1) x ($subgraph_num_vertices + 1);
my $pos = 0;
OUTER: for (;;) {
$used[$map[$pos]] = 0;
### undo use: "used=".join(',',@used)
for (;;) {
my $m = ++$map[$pos];
### $m
if ($m >= $num_vertices) {
$pos--;
### backtrack to pos: $pos
if ($pos < 0) {
return 0;
}
next OUTER;
}
if (! $used[$m]) {
$used[$m] = 1;
last;
}
### used ...
}
### incremented: "pos=$pos map=".join(',',@map)." used=".join(',',@used)
if ($degree[$map[$pos]] < $subgraph_degree[$pos]) {
### graph degree smaller than subgraph ...
next;
}
foreach my $p (0 .. $pos-1) {
### consider: "pos=$pos p=$p graph $map[$p] to $map[$pos] subgraph $p to $pos"
my $has_edge = ! $neighbour[$map[$p]][$map[$pos]];
my $subgraph_has_edge = ! $subgraph_neighbour[$p][$pos];
if ($has_edge != $subgraph_has_edge) {
next OUTER;
}
}
# good for this next vertex at $pos, descend
if (++$pos >= $subgraph_num_vertices) {
# print "found:\n";
# foreach my $p (0 .. $subgraph_num_vertices-1) {
# print " $p <-> $map[$p]\n";
# }
return join(', ', map {"$_=$map[$_]"} 0 .. $subgraph_num_vertices-1);
# return 1;
}
$map[$pos] = -1;
}
}
sub edge_aref_is_subgraph {
my ($edge_aref, $subgraph_edge_aref, %option) = @_;
### edge_aref_is_subgraph() ...
### $edge_aref
### $subgraph_edge_aref
my @degree;
my @neighbour;
foreach my $edge (@$edge_aref) {
$neighbour[$edge->[0]][$edge->[1]] = 1;
$neighbour[$edge->[1]][$edge->[0]] = 1;
$degree[$edge->[0]]++;
$degree[$edge->[1]]++;
}
### @degree
my @subgraph_degree;
my @subgraph_neighbour;
foreach my $edge (@$subgraph_edge_aref) {
$subgraph_neighbour[$edge->[0]][$edge->[1]] = 1;
$subgraph_neighbour[$edge->[1]][$edge->[0]] = 1;
$subgraph_degree[$edge->[0]]++;
$subgraph_degree[$edge->[1]]++;
}
### @subgraph_degree
if (defined (my $num_vertices = $option{'num_vertices'})) {
$num_vertices >= @degree
or croak "num_vertices option too small";
$#degree = $num_vertices - 1;
$#neighbour = $num_vertices - 1;
}
if (defined (my $subgraph_num_vertices = $option{'subgraph_num_vertices'})) {
$subgraph_num_vertices >= @subgraph_degree
or croak "num_vertices option too small";
$#subgraph_degree = $subgraph_num_vertices - 1;
$#subgraph_neighbour = $subgraph_num_vertices - 1;
}
if (@degree < @subgraph_degree) {
### graph fewer vertices than subgraph ...
return 0;
}
my $num_vertices = scalar(@neighbour);
my $subgraph_num_vertices = scalar(@subgraph_neighbour);
foreach my $i (0 .. $num_vertices-1) {
$degree[$i] ||= 0;
}
foreach my $i (0 .. $subgraph_num_vertices-1) {
$subgraph_degree[$i] ||= 0;
}
my @used = (0) x ($num_vertices + 1);
my @map = (-1) x ($subgraph_num_vertices + 1);
my $pos = 0;
OUTER: for (;;) {
$used[$map[$pos]] = 0;
### undo use: "used=".join(',',@used)
for (;;) {
my $m = ++$map[$pos];
### $m
if ($m >= $num_vertices) {
$pos--;
### backtrack to pos: $pos
if ($pos < 0) {
return 0;
}
next OUTER;
}
if (! $used[$m]) {
$used[$m] = 1;
last;
}
### used ...
}
### incremented: "pos=$pos map=".join(',',@map)." used=".join(',',@used)
if (($degree[$map[$pos]]||0) < ($subgraph_degree[$pos]||0)) {
### graph degree smaller than subgraph ...
### degree: $degree[$map[$pos]]
### subgraph degree: $subgraph_degree[$pos]
next;
}
foreach my $p (0 .. $pos-1) {
### consider: "pos=$pos p=$p graph $map[$p] to $map[$pos] subgraph $p to $pos"
my $has_edge = $neighbour[$map[$p]][$map[$pos]];
my $subgraph_has_edge = $subgraph_neighbour[$p][$pos];
if ($subgraph_has_edge && ! $has_edge) {
next OUTER;
}
}
# good for this next vertex at $pos, descend
if (++$pos >= $subgraph_num_vertices) {
# print "found:\n";
# foreach my $p (0 .. $subgraph_num_vertices-1) {
# print " $p <-> $map[$p]\n";
# }
return join(', ', map {"$_=$map[$_]"} 0 .. $subgraph_num_vertices-1);
# return 1;
}
$map[$pos] = -1;
}
}
sub edge_aref_degrees_allow_subgraph {
my ($edge_aref, $subgraph_edge_aref) = @_;
if (@$edge_aref < @$subgraph_edge_aref) {
return 0;
}
my @degree;
foreach my $edge (@$edge_aref) {
$degree[$edge->[0]]++;
$degree[$edge->[1]]++;
}
### @degree
my @subgraph_degree;
foreach my $edge (@$subgraph_edge_aref) {
$subgraph_degree[$edge->[0]]++;
$subgraph_degree[$edge->[1]]++;
}
### @subgraph_degree
@degree = sort {$b<=>$a} @degree; # descending
@subgraph_degree = sort {$b<=>$a} @subgraph_degree;
foreach my $i (0 .. $#subgraph_degree) {
if ($subgraph_degree[$i] > $degree[$i]) {
return 0;
}
}
return 1;
}
sub edge_aref_eccentricity {
my ($edge_aref, $v) = @_;
### $v
my $eccentricity = 0;
my @edges = @$edge_aref;
my @pending = ($v);
while (@pending) {
### @edges
### @pending
$eccentricity++;
my @new_pending;
foreach my $v (@pending) {
@edges = grep {
my ($from,$to) = @$_;
my $keep = 1;
if ($from == $v) {
push @new_pending, $to;
$keep = 0;
} elsif ($to == $v) {
push @new_pending, $from;
$keep = 0;
}
$keep
} @edges;
}
@pending = @new_pending;
}
return $eccentricity;
}
# return true if all vertices of $graph have same degree
sub edge_aref_is_regular {
my ($edge_aref) = @_;
my @degrees = edge_aref_degrees($edge_aref);
### @degrees
foreach my $i (1 .. $#degrees) {
if ($degrees[$i] != $degrees[0]) {
return 0;
}
}
return 1;
}
#------------------------------------------------------------------------------
sub Graph_Wiener_part_at_vertex {
my ($graph,$vertex) = @_;
my $total = 0;
$graph->for_shortest_paths(sub {
my ($t, $u,$v, $n) = @_;
if ($u eq $vertex) {
$total += $t->path_length($u,$v);
}
});
return $total;
}
sub Graph_Wiener_index {
my ($graph) = @_;
my $total = 0;
$graph->for_shortest_paths(sub {
my ($t, $u,$v, $n) = @_;
$total += $t->path_length($u,$v);
});
return $total/2;
}
sub Graph_terminal_Wiener_index {
my ($graph) = @_;
my $total = 0;
my $for = $graph->for_shortest_paths
(sub {
my ($t, $u,$v, $n) = @_;
### u: $graph->vertex_degree($u)
### v: $graph->vertex_degree($v)
if ($graph->vertex_degree($u) == 1 && $graph->vertex_degree($v) == 1) {
$total += $t->path_length($u,$v);
}
});
return $total/2;
}
sub Graph_terminal_Wiener_part_at_vertex {
my ($graph, $vertex) = @_;
### Graph_terminal_Wiener_part_at_vertex(): $vertex
my $total = 0;
my $for = $graph->for_shortest_paths
(sub {
my ($t, $u,$v, $n) = @_;
# ### u: $graph->vertex_degree($u)
# ### v: $graph->vertex_degree($v)
### path: "$u to $v"
# can have $vertex not a leaf node
if ($u eq $vertex
&& $graph->vertex_degree($v) == 1) {
### length: $t->path_length($u,$v)
$total += $t->path_length($u,$v);
}
});
return $total;
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm.
# Return a new Graph.pm which is its line graph.
# Each vertex of the line graph is an edge of $graph and edges in the line
# graph are between those $graph edges with a vertex in common.
#
sub Graph_line_graph {
my ($graph) = @_;
my $line = Graph->new (undirected => $graph->is_undirected);
$line->set_graph_attribute
(name => join(', ',
($graph->get_graph_attribute('name') // ()),
'line graph'));
foreach my $from_edge ($graph->edges) {
my $from_edge_name = join(':', @$from_edge);
my ($from_vertex, $to_vertex) = @$from_edge;
foreach my $to_edge ($graph->edges_at($from_vertex),
$graph->edges_at($to_vertex)) {
my $to_edge_name = join(':', @$to_edge);
if ($from_edge_name ne $to_edge_name
&& ! $line->has_edge ($from_edge_name, $to_edge_name)
&& ! $line->has_edge ($to_edge_name, $from_edge_name)) {
$line->add_edge($from_edge_name, $to_edge_name);
}
}
}
return $line;
}
sub Graph_Easy_line_graph {
my ($easy) = @_;
my $line = Graph::Easy->new (undirected => $easy->is_undirected);
foreach my $from_edge ($easy->edges) {
my $from_name = $from_edge->name;
foreach my $to_edge ($from_edge->from->edges,
$from_edge->to->edges) {
my $to_name = $to_edge->name;
if ($from_name ne $to_name
&& ! $line->has_edge ($from_name, $to_name)
&& ! $line->has_edge ($to_name, $from_name)) {
$line->add_edge($from_name, $to_name);
}
}
}
return $line;
}
# Graph_Beineke_graphs() returns a list of Graph.pm graphs of Beineke G1 to G9.
use constant::defer Graph_Beineke_graphs => sub {
require Graph::Maker::Beineke;
map {
Graph::Maker->new('Beineke', G=>$_, undirected=>1)
} 1 .. 9;
};
# $graph is a Graph.pm.
# Return true if $graph is a line graph, by checking none of Beineke G1 to
# G9 are induced subgraphs.
#
sub Graph_is_line_graph_by_Beineke {
my ($graph) = @_;
### Graph_is_line_graph_by_Beineke() ...
foreach my $G (Graph_Beineke_graphs()) {
if (Graph_is_induced_subgraph($graph, $G)) {
### is induced subgraph, so not line graph ...
return 0;
}
### not subgraph ...
}
### final is line graph ...
return 1;
}
#------------------------------------------------------------------------------
# Graph Doubles
# $graph is a Graph.pm.
# Return a new graph which is the bipartite double of $graph.
# The new graph is two copies of the original vertices "$v.A" and $v.B".
# An edge $u to $v in $graph becomes edges $u.A to $v.B
# and $u.B to $v.A
#
sub Graph_bipartite_double {
my ($graph) = @_;
my $double = $graph->new; # same directed, countedged, etc
foreach my $v ($graph->vertices) {
$double->add_vertex("$v.A");
$double->add_vertex("$v.B");
}
foreach my $edge ($graph->edges) {
my ($from,$to) = @$edge;
### edge: "$from to $to"
$double->add_edge("$from.A","$to.B");
$double->add_edge("$from.B","$to.A");
}
return $double;
}
#------------------------------------------------------------------------------
# GraphViz2 conversions
# file:///usr/share/doc/graphviz/html/info/attrs.html
# $graph is a Graph.pm object.
# Return a GraphViz2 object.
#
sub Graph_to_GraphViz2 {
my ($graph, %options) = @_;
### Graph_to_GraphViz2: %options
require GraphViz2;
$options{'vertex_name_type'}
//= $graph->get_graph_attribute('vertex_name_type') // '';
my $is_xy = ($options{'is_xy'}
|| $options{'vertex_name_type'} =~ /^xy/
|| $graph->get_graph_attribute('vertex_name_type_xy')
|| $graph->get_graph_attribute('vertex_name_type_xy_triangular'));
my $is_xy_triangular
= ($graph->get_graph_attribute('is_xy_triangular')
|| $options{'vertex_name_type'} =~ /^xy-triangular/
|| $graph->get_graph_attribute('vertex_name_type_xy_triangular'));
### $is_xy
### $is_xy_triangular
my $name = $graph->get_graph_attribute('name');
my $flow = ($options{'flow'} // $graph->get_graph_attribute('flow') // 'down');
if ($flow eq 'north') { $flow = 'BT'; }
if ($flow eq 'east') { $flow = 'LR'; }
my $graphviz2 = GraphViz2->new
(global => { directed => $graph->is_directed },
graph => { (defined $name ? (label => $name) : ()),
(defined $flow ? (rankdir => $flow) : ()),
# ENHANCE-ME: take this in %options somehow
# Scale like "3" means input coordinates are tripled, so
# actual drawing is 1/3 of an inch steps.
inputscale => 3,
},
node => { margin => 0, # cf default 0.11,0.055
},
);
foreach my $v ($graph->vertices) {
my @attrs;
if (my ($x,$y) = Graph_vertex_xy($graph,$v)) {
if ($is_xy_triangular) {
$y = sprintf '%.5f', $y*sqrt(3);
}
if (defined $options{'scale'}) {
$x *= $options{'scale'};
$y *= $options{'scale'};
}
push @attrs, pin=>1, pos=>"$x,$y";
### @attrs
}
if (defined(my $name = $graph->get_vertex_attribute($v,'name'))) {
push @attrs, label => $name;
}
$graphviz2->add_node(name => $v,
margin => '0.03,0.02', # cf default 0.11,0.055
height => '0.1', # inches, minimum
width => '0.1', # inches, minimum
@attrs);
}
foreach my $edge ($graph->edges) {
my ($from, $to) = @$edge;
$graphviz2->add_edge(from => $from, to => $to);
}
return $graphviz2;
}
sub Graph_vertex_xy {
my ($graph, $v) = @_;
if (defined (my $xy = $graph->get_vertex_attribute($v,'xy'))) {
return split /,/, $xy;
}
if ($graph->get_graph_attribute('vertex_name_type_xy_triangular')) {
my ($x,$y) = split /,/, $v;
return ($x, $y*sqrt(3));
}
if ($graph->get_graph_attribute('vertex_name_type_xy')) {
return split /,/, $v;
}
if (defined(my $x = $graph->get_vertex_attribute($v,'x'))
&& defined(my $y = $graph->get_vertex_attribute($v,'y'))) {
return ($x,$y);
}
return ();
}
sub Graph_set_xy_points {
my $graph = shift;
while (@_) {
my $v = shift;
my $point = shift;
### $v
### $point
$graph->set_vertex_attribute($v, x => $point->[0]);
$graph->set_vertex_attribute($v, y => $point->[1]);
}
}
# $graphviz2 is a GraphViz2 object.
#
sub GraphViz2_view {
my ($graphviz2, %options) = @_;
require File::Temp;
my $ps = File::Temp->new (UNLINK => 0, SUFFIX => '.ps');
my $ps_filename = $ps->filename;
$graphviz2->run(format => 'ps',
output_file => $ps_filename,
($options{'driver'} ? (driver => $options{'driver'}) : ()),
);
postscript_view_file($ps_filename, %options);
# $graphviz2->run(format => 'xlib',
# driver => 'neato',
# );
}
sub parent_aref_view {
my ($aref) = @_;
Graph_Easy_view(parent_aref_to_Graph_Easy($aref));
}
#------------------------------------------------------------------------------
# $name is a vertex name.
# Return a form suitable for use as a PGF/Tikz node name.
# ENHANCE-ME: not quite right, would want to fixup most parens and more too.
#
sub vertex_name_to_tikz {
my ($name) = @_;
$name =~ s/[,:]/-/g;
return $name;
}
# $graph is an undirected Graph.pm.
# Print some PGF/Tikz TeX nodes and edges.
# The output is a bit rough, and usually must be massaged by hand.
#
sub Graph_print_tikz {
my ($graph) = @_;
my $is_xy = $graph->get_graph_attribute('vertex_name_type_xy');
my @vertices = sort $graph->vertices;
my $flow = 'east';
my $rows = int(sqrt(scalar(@vertices)));
my $r = 0;
my $c = 0;
my %seen_vn;
foreach my $v (@vertices) {
my $x = ($flow eq 'west' ? -$c : $c);
my $vn = vertex_name_to_tikz($v);
if (exists $seen_vn{$vn}) {
croak "Oops, duplicate tikz vertex name for \"$v\" and \"$seen_vn{$vn}\"";
}
$seen_vn{$vn} = $v;
my $at = ($is_xy ? $v : "$x,$r");
print " \\node ($vn) at ($at) [my box] {$v};\n";
$r++;
if ($r >= $rows) {
$c++;
$r = 0;
}
}
print "\n";
my $arrow = $graph->is_directed ? "->" : "";
foreach my $edge ($graph->unique_edges) {
my ($from,$to) = @$edge;
my $count = $graph->get_edge_count($from,$to);
my $node = ($count == 1 ? ''
: "node[pos=.5,auto=left] {$count} ");
$from = vertex_name_to_tikz($from);
$to = vertex_name_to_tikz($to);
if ($from eq $to) {
print " \\draw [$arrow,loop below] ($from) to $node();\n";
} else {
print " \\draw [$arrow] ($from) to $node($to);\n";
}
}
print "\n";
}
sub all_looks_like_consecutive_number {
all_looks_like_number(@_) or return 0;
my @a = sort {$a<=>$b} @_;
foreach my $i (1 .. $#a) {
$a[$i] == $a[$i-1] + 1 or return 0;
}
return 1;
}
sub all_looks_like_number {
foreach (@_) {
(Scalar::Util::looks_like_number($_)
&& $_ <= (1<<24))
or return 0;
}
return 1;
}
sub sort_num_or_alnum {
foreach (@_) {
unless (Scalar::Util::looks_like_number($_)) {
return sort @_;
}
}
return sort {$a<=>$b} @_;
}
sub Graph_print_dreadnaut {
my ($graph) = @_;
print Graph_dreadnaut_str($graph);
}
sub Graph_dreadnaut_str {
my ($graph, %options) = @_;
my @vertices = $graph->vertices;
my $base;
if (@vertices && all_looks_like_number(@vertices)) {
### numeric vertices ...
@vertices = sort {$a<=>$b} @vertices;
$base = $vertices[0];
} else {
### non-numeric vertices, sort ...
@vertices = sort @vertices;
$base = $options{'base'} || 0;
}
my %vertex_to_n;
@vertex_to_n{@vertices} = $base .. $base+$#vertices; # hash slice
### %vertex_to_n
my $str = '';
$str .= ($graph->is_directed ? 'd' : '-d')
. ' n='.scalar(@vertices)
. " \$=$base g";
my $comma = '';
my $prev_i = 0;
my @edges = sort {$vertex_to_n{$a->[0]} <=> $vertex_to_n{$b->[0]}
|| $vertex_to_n{$a->[1]} <=> $vertex_to_n{$b->[1]}}
$graph->edges;
### num edges: scalar(@edges)
my $prev_from = $base;
my $join = '';
foreach my $edge (@edges) {
### $edge
$str .= $comma;
my $from = $vertex_to_n{$edge->[0]};
my $to = $vertex_to_n{$edge->[1]};
### indices: "$from to $to"
if ($from != $prev_from) {
$str .= ($from == $prev_from + 1 ? ';'
: "$join$from:");
$join = '';
$prev_from = $from;
}
$str .= "$join$to";
$join = ' ';
}
### $str
return $str . ".";
}
sub Graph_run_dreadnaut {
my ($graph, %options) = @_;
require IPC::Run;
my $str = Graph_dreadnaut_str($graph,%options) . " a x\n";
if ($options{'verbose'}) {
print $str;
}
if (! IPC::Run::run(['dreadnaut'], '<',\$str)) {
die "dreadnaut: $!";
}
}
#------------------------------------------------------------------------------
# $graph is an undirected Graph.pm.
# Return the clique number of $graph.
# The clique number is the number of vertices in the maximum clique
# (complete graph) contained in $graph.
# Currently this is a brute force search, so quite slow and suitable only for
# small number of vertices.
#
sub Graph_clique_number {
my ($graph) = @_;
my @vertices = sort $graph->vertices;
my @clique = (-1);
my $maximum_clique_size = 0;
my $pos = 0;
OUTER: for (;;) {
### at: join(',',@clique[0..$pos])
if (++$clique[$pos] > $#vertices) {
# backtrack
if (--$pos < 0) {
last;
}
next;
}
my $v = $vertices[$clique[$pos]];
foreach my $i (0 .. $pos-1) {
if (! $graph->has_edge($v, $vertices[$clique[$i]])) {
next OUTER;
}
}
$pos++;
if ($pos > $maximum_clique_size) {
# print " new high $maximum_clique_size\n";
$maximum_clique_size = $pos;
}
if ($pos > $#vertices) {
# $graph is a complete-N
last;
}
$clique[$pos] = $clique[$pos-1];
}
return $maximum_clique_size;
}
# $graph is a Graph.pm and @vertices are vertex names in it.
# Return true if those vertices are a clique, meaning edge between all pairs.
sub Graph_is_clique {
my ($graph, @vertices) = @_;
foreach my $i (0 .. $#vertices) {
$graph->has_vertex($vertices[$i]) or die;
foreach my $j (0 .. $#vertices) {
next if $i == $j;
### has: "$vertices[$i] $vertices[$j] is ".($graph->has_edge($vertices[$i], $vertices[$j])||0)
$graph->has_edge($vertices[$i], $vertices[$j]) or return 0;
}
}
return 1;
}
#------------------------------------------------------------------------------
# $graph is a tree
# $v is a child node of $parent
# Return the depth of the subtree $v and deeper underneath $parent.
# If $v is a leaf then it is the entire subtree and the return is depth 1.
#
sub Graph_subtree_depth {
my ($graph, $parent, $v) = @_;
### $parent
### $v
$graph->has_edge($parent,$v) or die "oops, $parent and $v not adjacent";
my $depth = 0;
my %seen = ($parent => 1, $v => 1);
my @pending = ($v);
do {
@pending = map {$graph->neighbours($_)} @pending;
@pending = grep {! $seen{$_}++} @pending;
$depth++;
} while (@pending);
return $depth;
}
# $graph is a tree
# $v is a child node of $parent
# return the children of $v, being all neighbours except $parent
sub Graph_subtree_children {
my ($graph, $parent, $v) = @_;
return grep {$_ ne $parent} $graph->neighbours($v);
}
#------------------------------------------------------------------------------
# $edge_aref is an arrayref [ [from,to], [from,to], ... ]
# where each vertex is integer 0 upwards
# Return the number of vertices, which means the maximum + 1 of the vertex
# numbers in the elements.
#
sub edge_aref_num_vertices {
my ($edge_aref) = @_;
if (! @$edge_aref) { return 0; }
return max(map {@$_} @$edge_aref) + 1;
}
# $edge_aref is an arrayref [ [from,to], [from,to], ... ]
# where each vertex is integer 0 upwards forming a tree with root 0
# Return an arrayref of the parent of each vertex, so $a->[i] = parent of i
#
sub edge_aref_to_parent_aref {
my ($edge_aref) = @_;
### edge_aref_to_parent_aref() ...
my @neighbours;
foreach my $edge (@$edge_aref) {
my ($from, $to) = @$edge;
push @{$neighbours[$from]}, $to;
push @{$neighbours[$to]}, $from;
}
my @parent;
my @n_to_v = (0);
my @v_to_n = (0);
my $upto_v = 1;
for (my $v = 0; $v < $upto_v; $v++) {
### neighbours: "$v=n$v_to_n[$v] to n=".join(',',@{$neighbours[$v_to_n[$v]]})
foreach my $n (@{$neighbours[$v_to_n[$v]]}) {
if (! defined $n_to_v[$n]) {
$n_to_v[$n] = $upto_v;
$v_to_n[$upto_v] = $n;
$parent[$upto_v] = $v;
$upto_v++;
}
}
}
foreach my $edge (@$edge_aref) {
foreach my $n (@$edge) {
$n = $n_to_v[$n]; # mutate array
}
}
### @parent
### num_vertices: scalar(@parent)
return \@parent;
}
# $parent_aref is an arrayref where $a->[i] = parent of i
# vertices are integers 0 upwards
# Return an edge aref [ [from,to], [from,to], ... ]
#
sub parent_aref_to_edge_aref {
my ($parent_aref) = @_;
return [ map {[$parent_aref->[$_] => $_]} 1 .. $#$parent_aref ];
}
# $parent_aref is an arrayref where $a->[i] = parent of i
# vertices are integers 0 upwards
# Return a Graph::Easy
#
sub parent_aref_to_Graph_Easy {
my ($parent_aref) = @_;
require Graph::Easy;
my $graph = Graph::Easy->new(undirected => 1);
if (@$parent_aref) {
$graph->add_vertex(0);
foreach my $v (1 .. $#$parent_aref) {
$graph->add_edge($v,$parent_aref->[$v]);
}
}
return $graph;
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm.
# Modify $graph by changing the name of vertex $old_name to $new_name.
# If $old_name and $new_name are the same then do nothing.
# Otherwise $new_name should not exist already.
#
sub Graph_rename_vertex {
my ($graph, $old_name, $new_name) = @_;
### $old_name
### $new_name
return if $old_name eq $new_name;
if ($graph->has_vertex($new_name)) {
croak "Graph vertex \"$new_name\" exists already";
}
$graph->add_vertex($new_name);
$graph->set_vertex_attributes($new_name,
$graph->get_vertex_attributes($old_name));
foreach my $edge ($graph->edges_at($old_name)) {
my ($from,$to) = @$edge;
if ($from eq $old_name) { $from = $new_name; }
if ($to eq $old_name) { $to = $new_name; }
### $from
### $to
$graph->add_edge($from,$to);
}
$graph->delete_vertex($old_name);
}
# $graph is a Graph.pm.
# Return a new vertex name for $graph, one which does not otherwise occur in
# $graph.
#
sub Graph_new_vertex_name {
my ($graph, $prefix) = @_;
if (! defined $prefix) { $prefix = ''; }
my $upto = $graph->get_graph_attribute('Graph_new_vertex_name_upto') // 0;
$upto++;
$graph->set_graph_attribute('Graph_new_vertex_name_upto',$upto);
return "$prefix$upto";
}
# $graph is a Graph.pm.
# Add vertices to pad out existing vertices to all degree $N.
sub Graph_pad_degree {
my ($graph, $N) = @_;
my $upto = 1;
my @original_vertices = $graph->vertices;
foreach my $v (@original_vertices) {
while ($graph->vertex_degree($v) < $N) {
$graph->add_edge($v, Graph_new_vertex_name($graph));
$graph->set_graph_attribute('vertex_name_type',undef);
}
}
return $graph;
}
# $graph is a Graph.pm.
sub Graph_degree_sequence {
my ($graph) = @_;
return sort {$a<=>$b} map {$graph->vertex_degree($_)} $graph->vertices;
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm.
# Replace each vertex by a star of N vertices.
# Existing edges become edges between an arm of the new stars.
# All vertices must be degree <= N-1 (the arms of the stars)
#
# Key/value options are
#
# edges_between => $integer, default 1
# Number of edges in connections between new stars.
# Default 1 is replacing each edge by an edge between the stars.
# > 1 means extra vertices for those connections.
# 0 means the stars have a vertex in common for existing edges.
#
sub Graph_star_replacement {
my ($graph, $N, %options) = @_;
my $new_graph = $graph->new (undirected => $graph->is_undirected);
my $edges_between = $options{'edges_between'} // 1;
### $edges_between
my $upto = 1;
my %v_to_arms;
foreach my $v ($graph->vertices) {
my $centre = $upto++;
foreach my $i (2 .. $N) {
my $arm = $upto++;
$new_graph->add_edge($centre,$arm);
push @{$v_to_arms{$v}}, $arm;
}
}
foreach my $edge ($graph->edges) {
my ($u,$v) = @$edge;
$u = (pop @{$v_to_arms{$u}}) // croak "oops, degree > $N";
$v = (pop @{$v_to_arms{$v}}) // croak "oops, degree > $N";
if ($edges_between == 0) {
Graph_merge_vertices($new_graph, $u, $v);
} else {
my @between = map {my $b = $upto++; $b} 2 .. $edges_between;
$new_graph->add_path($u, @between, $v);
}
}
if (defined (my $name = $graph->get_graph_attribute('name'))) {
my $append = ", $N-star rep";
if ($name =~ /\Q$append\E$/) { $name .= ' 2'; }
elsif ($name =~ s{(\Q$append\E )(\d+)$}{$1.($2+1)}e) { }
else { $name .= $append; }
$graph->set_graph_attribute (name => $name);
### $name
}
return $new_graph;
}
sub _closest_xy_pair {
my ($aref, $bref) = @_;
if (@$aref == 0 || @$bref == 0) { return; }
my $min_a = 0;
my $min_b = 0;
my $min_norm;
foreach my $a (0 .. $#$aref) {
my ($ax,$ay) = split /,/, $aref->[$a];
foreach my $b (0 .. $#$bref) {
my ($bx,$by) = split /,/, $bref->[$b];
my $norm = ($ax-$bx)**2 + ($ay-$by)**2;
if (! defined $min_norm || $norm < $min_norm) {
$min_a = $a;
$min_b = $b;
$min_norm = $norm;
}
}
}
return (splice(@$aref, $min_a, 1),
splice(@$bref, $min_b, 1));
}
# Graph_merge_vertices($graph, $v, $v2, $v3, ...)
# $graph is a Graph.pm
# Modify $graph to merge all the given vertices into one.
# Edges going to any of them are moved to go to $v, and the rest deleted.
# Only for undirected graphs currently.
#
sub Graph_merge_vertices {
my $graph = shift;
$graph->expect_undirected;
my $v = shift;
foreach my $other (@_) {
### Graph_merge_vertices(): "$v, $other"
foreach my $neighbour ($graph->neighbours($other)) {
### $neighbour
unless ($neighbour eq $v) {
$graph->add_edge ($v, $neighbour);
}
}
$graph->delete_vertex($other);
}
}
# $graph is a Graph.pm.
# Replace each vertex by an N-cycle.
# Existing edges become edges between vertices of the cycles, consecutively
# around the cycle.
#
sub Graph_cycle_replacement {
my ($graph, $N, %options) = @_;
my $edges_between = $options{'edges_between'} // 1;
my $vertex_name_type = $graph->get_graph_attribute('vertex_name_type') // '';
my $xy = ($vertex_name_type =~ /^xy/) && $N==4;
### $vertex_name_type
### $xy
my $new_graph = $graph->new (undirected => $graph->is_undirected);
my $upto = 1;
my %v_to_arms;
foreach my $v ($graph->vertices) {
my @c;
if ($xy) {
my ($x,$y) = split /,/,$v;
$x *= $edges_between+4;
$y *= $edges_between+4;
@c = ( ($x+1).','.($y+1),
($x-1).','.($y+1),
($x-1).','.($y-1),
($x+1).','.($y-1) );
} else {
@c = map {my $c = $upto++; $c} 1 .. $N;
}
foreach my $c (@c) { die if $new_graph->has_vertex($c); }
$new_graph->add_cycle(@c);
$v_to_arms{$v} = \@c;
}
foreach my $edge ($graph->edges) {
my ($u,$v) = @$edge;
my @between;
if ($xy) {
($u,$v) = _closest_xy_pair($v_to_arms{$u},
$v_to_arms{$v})
or croak "oops, degree > $N";
# $u = (pop @{$v_to_arms{$u}}) // croak "oops, degree > $N";
# $v = (pop @{$v_to_arms{$v}}) // croak "oops, degree > $N";
my ($ux,$uy) = split /,/,$u;
my ($vx,$vy) = split /,/,$v;
@between = map { my $x = $ux + ($vx-$ux)/($edges_between+2);
my $y = $uy + ($vy-$uy)/($edges_between+2);
my $b = "$x,$y";
die if $new_graph->has_vertex($b);
$b;
} 1 .. $edges_between;
} else {
$u = (shift @{$v_to_arms{$u}}) // croak "oops, degree > $N";
$v = (shift @{$v_to_arms{$v}}) // croak "oops, degree > $N";
@between = map {my $b = $upto++; $b} 1 .. $edges_between;
}
if ($edges_between == 0) {
Graph_merge_vertices($new_graph, $u, $v);
} else {
$new_graph->add_path($u, @between, $v);
}
}
if (defined (my $name = $graph->get_graph_attribute('name'))) {
my $append = ", $N-star rep";
if ($name =~ /\Q$append\E$/) {
$name .= ' 2';
} elsif ($name =~ s{(\Q$append\E )(\d+)$}{$1.($2+1)}e) {
} else {
$name .= $append;
}
$new_graph->set_graph_attribute (name => $name);
}
$new_graph->set_graph_attribute('vertex_name_type', $vertex_name_type);
return $new_graph;
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm.
# Return a list of vertices which are a path achieving the eccentricity of $u.
#
# FIXME: is $graph->longest_path args ($u,$v) a documented feature?
sub Graph_eccentricity_path {
my ($graph, $u) = @_;
$graph->expect_undirected;
my $max = 0;
my $max_v;
for my $v ($graph->vertices) {
next if $u eq $v;
my $len = $graph->path_length($u, $v);
if (defined $len && (! defined $max || $len > $max)) {
$max = $len;
$max_v = $v;
}
}
return $graph->longest_path($u,$max_v);
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm undirected tree.
# Return ($eccentricity, $vertex,$vertex) which is the centre 1 or 2 vertex
# names and their eccentricity.
# Only tested on bicentral trees.
# FIXME: the return is not eccentricity but num vertices to reach maximum?
sub Graph_tree_centre_vertices {
my ($graph) = @_;
{
my $eccentricity = 0;
my %seen;
my %unseen = map {$_=>1} $graph->vertices;
my @prev_unseen;
for (;;) {
### seen: join(' ',keys %seen)
### unseen: join(' ',keys %unseen)
### $eccentricity
%unseen or last;
$eccentricity++;
@prev_unseen = keys %unseen;
my @leaves;
foreach my $v (@prev_unseen) {
my @neighbours = grep {! exists $seen{$_}} $graph->neighbours($v);
if (@neighbours <= 1) {
push @leaves, $v;
}
}
### @leaves
delete @unseen{@leaves}; # leaf nodes go from unseen to seen
@seen{@leaves} = ();
}
return ($eccentricity, @prev_unseen);
}
{
$graph = $graph->copy;
my @prev_vertices;
for (;;) {
my @vertices = $graph->vertices
or last;
@prev_vertices = @vertices;
my @leaves = grep {$graph->degree($_) <= 1} @vertices;
$graph->delete_vertices(@leaves);
}
return @prev_vertices;
}
}
# $graph is an undirected connected Graph.pm.
# Return a list of its leaf vertices.
#
sub Graph_leaf_vertices {
my ($graph) = @_;
return grep {$graph->vertex_degree($_)<=1} $graph->vertices;
}
# $graph is a Graph.pm undirected tree.
# Return a list of vertices which attain the diameter of tree $graph.
#
sub Graph_tree_diameter_path {
my ($graph) = @_;
if ($graph->vertices == 0) { return; }
my ($eccentricity, @centres) = Graph_tree_centre_vertices($graph);
### @centres
my @paths = ([ $centres[0] ]);
my @prev_paths = @paths;
for (;;) {
### paths: map {join(',',@$_)} @paths
my @new_paths;
foreach my $path (@paths) {
my $v = $path->[-1];
foreach my $neighbour ($graph->neighbours($v)) {
next if @$path>=2 && $neighbour eq $path->[-2];
push @new_paths, [@$path,$neighbour];
}
}
if (@new_paths) {
@prev_paths = @paths;
@paths = @new_paths;
} else {
last;
}
}
my $path = shift @paths;
### final path: join(',',@$path)
push @paths, @prev_paths;
if (@paths) {
foreach my $other_path (@paths) {
### final path: join(',',@$path)
### consider other: join(',',@$other_path)
if (@$other_path < 2 || $other_path->[1] ne $path->[1]) {
my @join = reverse @$path;
pop @join;
push @join, @$other_path;
### join to: join(',',@join)
$path = \@join;
last;
}
}
}
### $eccentricity
### path length: scalar(@$path)
scalar(@$path) == 2*$eccentricity - (@centres==1)
or die "oops";
return @$path;
}
# $graph is an undirected connected Graph.pm.
# Return the number of paths attaining the diameter of $graph.
# A path u--v is counted just once, not also v--u.
#
sub Graph_diameter_count {
my ($graph) = @_;
if ($graph->vertices <= 1) {
return 1;
}
my $diameter = 0;
my $count = 0;
$graph->for_shortest_paths(sub {
my ($t, $u,$v, $n) = @_;
my $len = $t->path_length($u,$v);
if ($len > $diameter) {
### new high path length: $len
$count = 0;
$diameter = $len;
}
if ($len == $diameter) {
$count++;
### equal high path length to count: $count
}
});
### $diameter
return ($graph->is_undirected ? $count/2 : $count);
}
# $graph is a Graph.pm.
# Insert $n new vertices into each of its edges.
# If $n omitted or undef then default 1 vertex in each edge.
sub Graph_subdivide {
my ($graph, $n) = @_;
if (! defined $n) { $n = 1; }
foreach my $edge ($graph->edges) {
$graph->delete_edge (@$edge);
my $prefix = "$edge->[0]-$edge->[1]-";
$graph->add_path ($edge->[0],
(map {Graph_new_vertex_name($graph,$prefix)} 1 .. $n),
$edge->[1]);
}
if ($n && $graph->edges
&& defined (my $name = $graph->get_graph_attribute('name'))) {
$graph->set_graph_attribute (name =>
"$name subdivision".($n > 1 ? " $n" : ""));
}
return $graph;
}
#------------------------------------------------------------------------------
# Independence Number
# $graph is a Graph.pm undirected tree or forest.
# Return its independence number.
#
sub Graph_tree_indnum {
my ($graph) = @_;
### Graph_tree_indnum: "num_vertices ".scalar($graph->vertices)
$graph->expect_acyclic;
$graph = $graph->copy;
my $indnum = 0;
my %exclude;
OUTER: while ($graph->vertices) {
foreach my $v ($graph->vertices) {
my $degree = $graph->vertex_degree($v);
next unless $degree <= 1;
my ($u) = $graph->neighbours($v);
### consider: "$v degree $degree neighbours ".($u//'undef')
if (delete $exclude{$v}) {
### exclude ...
} else {
### leaf include ...
$indnum++;
if (defined $u) { $exclude{$u} = 1; }
}
$graph->delete_vertex($v);
next OUTER;
}
die "oops, not a tree";
}
return $indnum;
}
sub Graph_make_most_indomsets {
my ($n) = @_;
my $graph = Graph->new (undirected=>1);
my $v = 0;
while ($n > 0) {
if ($v) { $graph->add_edge(0,$v) } # to x
my $u = $v;
my $size = 3 + (($n%3)!=0);
foreach my $i (0 .. $size-1) { # triangle or complete-4
foreach my $j (0 .. $i-1) {
$graph->add_edge($u+$i, $u+$j);
}
}
$n -= $size;
$v += $size;
}
return $graph;
}
sub Graph_is_indset {
my ($graph,$aref) = @_;
foreach my $from (@$aref) {
foreach my $to (@$aref) {
if ($graph->has_edge($from,$to)) {
return 0;
}
}
}
return 1;
}
sub Graph_indnum_and_count {
my ($graph) = @_;
require Algorithm::ChooseSubsets;
my @vertices = sort $graph->vertices;
my $it = Algorithm::ChooseSubsets->new(\@vertices);
my $indnum = 0;
my $count = 0;
while (my $aref = $it->next) {
if (Graph_is_indset($graph,$aref)) {
if (@$aref == $indnum) {
$count++;
} elsif (@$aref > $indnum) {
$indnum = @$aref;
$count = 1;
}
}
}
return ($indnum, $count);
}
#------------------------------------------------------------------------------
# Domination Number
# Cockayne, Goodman, Hedetniemi, "A Linear Algorithm for the Domination
# Number of a Tree", Information Processing Letters, volume 4, number 2,
# November 1975, pages 41-44.
# $graph is a Graph.pm undirected tree or forest.
# Return its domination number.
#
sub Graph_tree_domnum {
my ($graph) = @_;
### Graph_tree_domnum: "num_vertices ".scalar($graph->vertices)
$graph->expect_acyclic;
$graph = $graph->copy;
my $domnum = 0;
my %mtype = map {$_=>'bound'} $graph->vertices;
OUTER: while ($graph->vertices) {
foreach my $v ($graph->vertices) {
my $degree = $graph->vertex_degree($v);
next unless $degree <= 1;
### consider: $v
### $degree
my ($u) = $graph->neighbours($v);
if ($mtype{$v} eq 'free') {
### free, delete ...
} elsif ($mtype{$v} eq 'bound') {
### bound ...
if (defined $u) {
### set neighbour $u required ...
$mtype{$u} = 'required';
} else {
### no neighbour, domnum++ ...
$domnum++;
}
} elsif ($mtype{$v} eq 'required') {
### required, domnum++ ...
$domnum++;
if (defined $u && $mtype{$u} eq 'bound') {
### set neighbour $u free ...
$mtype{$u} = 'free';
}
} else {
die;
}
delete $mtype{$v};
$graph->delete_vertex($v);
next OUTER;
}
die "oops, not a tree";
}
return $domnum;
}
#------------------------------------------------------------------------------
# Dominating Sets Count
# with(n) = prod(child any) # sets including parent
# undom(n) = prod(child dom) # sets without parent and parent undominated
# dom(n) = prod(child with + dom) - prod(child dom)
# # sets without parent and parent dominated
# T(n) = with(n) + dom(n);
#
# with + dom = any - undom
#
# *
# / | \
# * * *
# /|\ /|\ /|\
# * * * * * * * * *
#
# path 1--2 2 with + 1 without = 3 any 0 without undom
# path 1--2--3 3 with + 2 without = 5 any 1 without undom
# 2 without dom
# cannot e,1,3
# $graph is a Graph.pm tree.
# Return the number of dominating sets in $graph.
#
sub Graph_tree_domsets_count {
my ($graph) = @_;
require Math::BigInt;
$graph = $graph->copy;
$graph->vertices || return 1; # empty graph
my %data;
my $one = Math::BigInt->new(1);
OUTER: for (;;) {
foreach my $v (sort $graph->vertices) {
my $degree = $graph->vertex_degree($v);
next unless $degree <= 1;
# with(n) = prod(c any)
# without(n) = prod(c with + dom = domsets)
# without_undom(n) = prod(c dom);
#
my $c_with = $data{$v}->{'with'} // $one;
my $c_without_undom = $data{$v}->{'without_undom'} // $one;
my $c_without = $data{$v}->{'without'} // $one;
my $c_without_dom = $c_without - $c_without_undom;
my $c_domsets = $c_with + $c_without_dom;
my $c_any = $c_with + $c_without;
### consider: "$v deg=$degree with $c_with, without $c_without, without_undom $c_without_undom"
### consider: " so without_dom=$c_without_dom domsets=$c_domsets any=$c_any"
if ($degree == 0) {
return $c_domsets;
}
my ($u) = $graph->neighbours($v);
$data{$u}->{'with'} //= $one;
$data{$u}->{'without'} //= $one;
$data{$u}->{'without_undom'} //= $one;
$data{$u}->{'with'} *= $c_any;
$data{$u}->{'without'} *= $c_domsets;
$data{$u}->{'without_undom'} *= $c_without_dom;
delete $data{$v};
$graph->delete_vertex($v);
next OUTER;
}
die "oops, not a tree $graph";
}
# OUTER: for (;;) {
# foreach my $v (sort $graph->vertices) {
# my $degree = $graph->vertex_degree($v);
# next unless $degree <= 1;
#
# $data{$v}->{'prod_c_any'} //= $one;
# $data{$v}->{'prod_c_dom'} //= $one;
# $data{$v}->{'prod_c_with_or_dom'} //= $one;
#
# # with(n) = prod(c any)
# # undom(n) = prod(c dom);
# # dom(n) = prod(c with + dom) - prod(c dom)
# #
# my $with = $data{$v}->{'prod_c_any'};
# my $undom = $data{$v}->{'prod_c_dom'};
# my $dom = $data{$v}->{'prod_c_with_or_dom'} - $undom;
# my $ret = $with + $dom;
# my $any = $ret + $undom;
#
# ### consider: "$v deg=$degree prods $data{$v}->{'prod_c_any'}, $data{$v}->{'prod_c_dom'}, $data{$v}->{'prod_c_with_or_dom'}"
# ### consider: " with $with dom $dom undom=$undom, ret $ret any $any"
#
# if ($degree == 0) {
# return $ret;
# }
#
# my ($u) = $graph->neighbours($v);
# $data{$u}->{'prod_c_any'} //= $one;
# $data{$u}->{'prod_c_dom'} //= $one;
# $data{$u}->{'prod_c_with_or_dom'} //= $one;
# $data{$u}->{'prod_c_any'} *= $any;
# $data{$u}->{'prod_c_dom'} *= $dom;
# $data{$u}->{'prod_c_with_or_dom'} *= $ret;
#
# delete $data{$v};
# $graph->delete_vertex($v);
# next OUTER;
# }
# die "oops, not a tree $graph";
# }
}
# 1 2 3 4 5
# path 1,1,2,2,4,4,7,9,13,18,25,36,49
# 1 with=1=1+0 without=0=0+1 domsets=1+0 = 1
# 2 with=1=1+0 without=1=1+0 domsets=1+1 = 2
# 3 with=2=1+1 without=2=1+1 domsets=1+1 = 2
# 4 with=2=1+1 without=3=2+1 domsets=2+2 = 4
# 5 with=4=1+3 without=4=2+2 domsets=4+2 = 6
#
# 1,3,4 without_dom
# 2,4 without_dom
# 1,3,5 with_unreq
# 2,5 with_req
# 1,3,4,5 not minimal
# 2,4,5 with_unreq but itself not minimal
#
# $graph is a Graph.pm tree.
# Return the number of minimal dominating sets in $graph.
#
sub Graph_tree_minimal_domsets_count {
my ($graph) = @_;
return tree_minimal_domsets_count_data_ret
(Graph_tree_minimal_domsets_count_data($graph));
}
# $graph is a Graph.pm tree.
# Return a hashref of data counting minimal dominating sets in $graph.
#
sub Graph_tree_minimal_domsets_count_data {
my ($graph) = @_;
require Math::BigInt;
$graph->vertices
|| return tree_minimal_domsets_count_data_initial(); # empty graph
$graph = $graph->copy;
my %data;
foreach my $v ($graph->vertices) {
$data{$v} = tree_minimal_domsets_count_data_initial();
}
OUTER: for (;;) {
foreach my $c (sort {$a cmp $b} $graph->vertices) {
my $degree = $graph->vertex_degree($c);
next unless $degree <= 1;
my ($v) = $graph->neighbours($c)
or return $data{$c}; # root
$data{$v} //= tree_minimal_domsets_count_data_initial();
tree_minimal_domsets_count_data_product_into
($data{$v},
delete($data{$c}) // tree_minimal_domsets_count_data_initial());
$graph->delete_vertex($c);
next OUTER;
}
die "oops, not a tree $graph";
}
}
sub tree_minimal_domsets_count_data_initial {
my $zero = Math::BigInt->new(0);
my $one = Math::BigInt->new(1);
$zero = 0;
$one = 1;
return { with => $one,
with_notreq => $one,
with_min_notreq => $one,
without_dom_sole => $zero,
without_notsole => $one,
without_undom => $one,
};
}
sub tree_minimal_domsets_count_data_ret {
my ($data) = @_;
return ($data ->{'with'}
- $data->{'with_notreq'}
+ $data->{'with_min_notreq'}
+ $data->{'without_dom_sole'}
+ $data->{'without_notsole'}
- $data->{'without_undom'});
}
# The args are 0 or more tree_minimal_domsets hashrefs.
# Return their product. This is a tree_minimal_domsets hashref for a
# vertex which has the given args as child vertices.
#
sub tree_minimal_domsets_count_data_product {
return tree_minimal_domsets_count_data_product_into
(tree_minimal_domsets_count_data_initial(), @_);
}
# $p is a tree_minimal_domsets hashref and zero or more further args likewise
# which are children of $p.
# Return their product for $p with those children.
#
sub tree_minimal_domsets_count_data_product_into {
### tree_minimal_domsets_count_data_product_into() ...
my $p = shift;
### $p
foreach my $v (@_) {
# ### $v
my $v_with_notmin_notreq = $v->{'with_notreq'} - $v->{'with_min_notreq'};
my $v_without_dom_notsole = $v->{'without_notsole'} - $v->{'without_undom'};
my $v_without_dom = $v->{'without_dom_sole'} + $v_without_dom_notsole;
my $v_mindom = ($v->{'with'} - $v_with_notmin_notreq # with_min
+ $v_without_dom);
my $v_with_req = $v->{'with'} - $v->{'with_notreq'};
$p->{'with'} *= $v_with_req + $v->{'without_notsole'};
$p->{'with_notreq'} *= $v_with_req + $v_without_dom_notsole;
$p->{'with_min_notreq'} *= $v_without_dom_notsole;
$p->{'without_dom_sole'} = ($p->{'without_dom_sole'} * $v_without_dom
+ $p->{'without_undom'} * $v_with_notmin_notreq);
$p->{'without_notsole'} *= $v_mindom;
$p->{'without_undom'} *= $v_without_dom;
}
return $p;
}
# $graph is a Graph.pm.
# $aref is an arrayref of vertex names.
# Return true if these vertices are a dominating set in $graph.
#
sub Graph_is_domset {
my ($graph, $aref) = @_;
my %vertices; @vertices{$graph->vertices} = ();
delete @vertices{@$aref,
map {$graph->neighbours($_)} @$aref};
return keys(%vertices) == 0;
}
# $graph is a Graph.pm.
# $aref is an arrayref of vertex names.
# Return true if these vertices are minimal for the amount of $graph they
# dominate, meaning any vertex removed would reduce the amount of $graph
# dominated.
#
sub Graph_domset_is_minimal {
my ($graph, $aref) = @_;
my %count;
foreach my $v (@$aref) {
foreach my $d ($v, $graph->neighbours($v)) {
$count{$d}++;
}
}
V: foreach my $v (@$aref) {
foreach my $d ($v, $graph->neighbours($v)) {
if ($count{$d} < 2) { next V; }
}
return 0; # $v and neighbours all count >=2
}
return 1;
}
# $graph is a Graph.pm.
# $aref is an arrayref of vertex names.
# Return true if these vertices are a minimal dominating set in $graph.
#
sub Graph_is_minimal_domset {
my ($graph, $aref) = @_;
return Graph_is_domset($graph,$aref) && Graph_domset_is_minimal($graph,$aref);
}
# Return the number of minimal dominating sets in $graph by iterating
# through all vertex sets and testing by the Graph_is_minimal_domset()
# predicate. This is quite slow so suitable only for small number of
# vertices.
#
sub Graph_minimal_domsets_count_by_pred {
my ($graph) = @_;
return Graph_sets_count_by_pred($graph, \&Graph_is_minimal_domset);
}
sub Graph_sets_count_by_pred {
my ($graph, $func) = @_;
require Algorithm::ChooseSubsets;
my $count = 0;
my @vertices = sort $graph->vertices;
my $it = Algorithm::ChooseSubsets->new(\@vertices);
while (my $aref = $it->next) {
if ($func->($graph,$aref)) {
$count++;
}
}
return $count;
}
sub Graph_sets_minimum_and_count_by_pred {
my ($graph, $func) = @_;
require Algorithm::ChooseSubsets;
my @count;
my $minsize = $graph->vertices;
my @vertices = sort $graph->vertices;
my $it = Algorithm::ChooseSubsets->new(\@vertices);
while (my $aref = $it->next) {
my $size = @$aref;
next if $size > $minsize;
if ($func->($graph,$aref)) {
$count[$size]++;
$minsize = min($minsize,$size);
}
}
return ($minsize, $count[$minsize]);
}
#------------------------------------------------------------------------------
# Total Dominating Sets
# $graph is a Graph.pm.
# $aref is an arrayref of vertex names.
# Return true if these vertices are a total dominating set in $graph.
# Every vertex must have one of $aref as a neighbour.
# Unlike a plain dominating set, $aref vertices to not dominate themselves,
# they must have a neighbour in the set.
#
sub Graph_is_total_domset {
my ($graph, $aref) = @_;
my %vertices; @vertices{$graph->vertices} = ();
delete @vertices{map {$graph->neighbours($_)} @$aref};
return keys(%vertices) == 0;
}
#------------------------------------------------------------------------------
# $graph is a Graph.pm and $sptg is its $graph->SPT_Dijkstra() tree.
# Set $sptg vertex attribute "count" on each vertex $v which gives the count
# of number of paths from SPT_Dijkstra_root to that $v.
#
sub Graph_SPT_counts {
my ($graph,$sptg, %options) = @_;
my $start = $sptg->get_graph_attribute('SPT_Dijkstra_root');
my $one = $options{'one'} || 1;
$sptg ->set_vertex_attribute ($start,'count',$one);
foreach my $from (sort {($sptg->get_vertex_attribute($a,'weight') || 0)
<=>
($sptg->get_vertex_attribute($b,'weight') || 0)}
$sptg->vertices) {
my $target_distance
= ($sptg->get_vertex_attribute($from,'weight') || 0) + 1;
my $from_count = $sptg ->get_vertex_attribute($from,'count');
### from: $from . ' weight ' .($sptg->get_vertex_attribute($from,'weight') || 0)
### $from_count
### $target_distance
foreach my $to ($graph->neighbours($from)) {
if (($sptg->get_vertex_attribute($to,'weight') || 0)
== $target_distance) {
### to: $to . ' weight ' .($sptg->get_vertex_attribute($to,'weight') || 0)
$sptg ->set_vertex_attribute
($to,'count',
$from_count + ($sptg ->get_vertex_attribute($to,'count') || 0));
} else {
### skip: $to . ' weight ' .($sptg->get_vertex_attribute($to,'weight') || 0)
}
}
}
}
#------------------------------------------------------------------------------
# Cycles
sub Graph_is_cycle {
my ($graph, $aref) = @_;
foreach my $i (0 .. $#$aref) {
$graph->has_edge($aref->[$i], $aref->[$i-1]) or return 0;
}
return 1;
}
# $graph is a Graph.pm. Find all cycles in it.
# The return is a list of arrayrefs, with each arrayref containing vertices
# which are a cycle.
# Each cycle appears just once, so just one direction around, not both ways.
#
# The order of vertices within each cycle and the order of cycles in the
# return are both unspecified. Within each cycle has a canonical order, but
# don't rely on that. The order of cycles is hash-random.
#
sub Graph_find_all_cycles {
my ($graph) = @_;
my @paths = map {[$_]} $graph->vertices;
my @cycles;
while (@paths) {
### num paths: scalar @paths
my @new_paths;
foreach my $path (@paths) {
NEIGHBOUR: foreach my $next ($graph->neighbours($path->[-1])) {
next if $next lt $path->[0]; # must have start smallest
if ($next eq $path->[0]) { # back to start, len=1 or >=3
Graph_is_cycle($graph, $path) or die;
if (@$path!=2
&& $path->[1] lt $path->[-1]) { # direction smaller second only
push @cycles, $path;
}
} else {
foreach my $i (1 .. $#$path) {
next NEIGHBOUR if $next eq $path->[$i]; # back to non-start
}
push @new_paths, [ @$path, $next ];
}
}
}
@paths = @new_paths;
}
return @cycles;
}
sub Graph_num_cycles {
my ($graph) = @_;
my @cycles = Graph_find_all_cycles($graph);
return scalar @cycles;
}
# Return true if $graph has a bi-cyclic component, meaning a connected
# component with 2 or more cycles in it.
sub Graph_has_bicyclic_component {
my ($graph) = @_;
my @components = $graph->connected_components;
foreach my $component (@components) {
my $subgraph = $graph->subgraph($component);
if (MyGraphs::Graph_num_cycles($subgraph) >= 2) {
return 1;
}
}
return 0;
}
# length of the smallest cycle in $graph
sub Graph_girth {
my ($graph) = @_;
### Graph_girth() ...
my $num_vertices = scalar $graph->vertices;
my $girth;
my $min = $graph->is_directed ? 1 : 3;
OUTER: foreach my $from ($graph->vertices) {
### $from
my %seen = ($from => 1);
my @pending = ($from);
foreach my $len (1 .. ($girth||$num_vertices)) {
### at: "len=$len pending=".join(' ',@pending)
my @new_pending;
foreach my $to (map {$graph->successors($_)} @pending) {
if ($len>=$min && $to eq $from) {
### cycle: "to=$to len=$len"
if (!defined $girth || $len < $girth) {
### is new low ...
$girth = $len;
}
next OUTER;
}
unless ($seen{$to}++) {
push @new_pending, $to;
}
}
@pending = @new_pending;
}
}
return $girth;
}
# $graph is an undirected Graph.pm.
# If $v is in a hanging cycle, other than the attachment point, then return
# an arrayref of the vertices of that cycle other than the attachment point
# (in an unspecified order).
# For example,
#
# 4---5
# \ /
# 1---2---3---6
#
# has hanging cycle 3,4,5. $v=4 or $v=5 gives return is [4,5].
# If $v is not in a hanging cycle then return undef.
#
sub Graph_is_hanging_cycle {
my ($graph, $v) = @_;
if ($graph->degree($v) != 2) { return undef; }
my %cycle = ($v => 1);
my @pending = $graph->neighbours($v);
my @end;
while (@pending) {
$v = pop @pending;
next if $cycle{$v};
if ($graph->degree($v) != 2) {
push @end, $v;
next;
}
$cycle{$v} = 1;
push @pending, $graph->neighbours($v);
}
if (@end == 0 || (@end==2 && $end[0] eq $end[1])) {
return [ keys %cycle ];
} else {
return undef;
}
}
# $graph is an undirected Graph.pm.
# Modify $graph to remove any hanging cycles.
# For example,
#
# 4---5
# \ /
# 1---2---3---6
#
# has hanging cycle 3,4,5. Vertices 4,5 are removed.
#
sub Graph_delete_hanging_cycles {
my ($graph) = @_;
my $count = 0;
MORE: for (;;) {
foreach my $v ($graph->vertices) {
if (my $aref = Graph_is_hanging_cycle($graph,$v)) {
$graph->delete_vertices(@$aref);
$count++;
next MORE;
}
}
last;
}
if ($count
&& defined(my $name = $graph->get_graph_attribute('name'))) {
$graph->set_graph_attribute (name => "$name, stripped hanging");
}
return $count;
}
# d-----c
# | |
# a-----b
sub Graph_find_all_4cycles {
my ($graph, %options) = @_;
### Graph_find_all_4cycles() ...
my $callback = $options{'callback'} || sub{};
my %seen;
foreach my $a (sort $graph->vertices) {
my @a_neighbours = $graph->neighbours($a);
### a: "$a to ".join(',',@a_neighbours)
foreach my $b (@a_neighbours) {
next if $b eq $a; # ignore self-loops
my @b_neighbours = $graph->neighbours($b);
if (! $graph->has_edge($a,$b)) {
print " a=$a\n";
foreach my $neighbour (@a_neighbours) {
print " $neighbour\n";
}
die "oops, no edge $a to $b";
}
foreach my $c (@b_neighbours) {
next if $c eq $a;
next if $c eq $b;
my @c_neighbours = $graph->neighbours($c);
if (! $graph->has_edge($b,$c)) {
die "oops";
}
foreach my $d (@c_neighbours) {
if (! $graph->has_edge($c,$d)) {
die "oops";
}
next if $d eq $a;
next if $d eq $b;
next if $d eq $c;
my @d_neighbours = $graph->neighbours($d);
### $d
### cycle: "$a $b $c $d goes ".join(',',@d_neighbours)
next unless $graph->has_edge($d,$a) || $graph->has_edge($a,$d);
next if $seen{$a,$b,$c,$d}++;
next if $seen{$b,$c,$d,$a}++;
next if $seen{$c,$d,$a,$b}++;
next if $seen{$d,$a,$b,$c}++;
next if $seen{$d,$c,$b,$a}++;
next if $seen{$c,$b,$a,$d}++;
next if $seen{$b,$a,$d,$c}++;
next if $seen{$a,$d,$c,$b}++;
# print "raw ",join(' -- ',($a,$b,$c,$d)),"\n";
# print " has_edge ",$graph->has_edge($a,$b),"\n";
# print " has_edge ",$graph->has_edge($b,$c),"\n";
# print " has_edge ",$graph->has_edge($c,$d),"\n";
# print " has_edge ad ",$graph->has_edge($d,$a),"\n";
# must not mutate the loop variables $a,$b,$c,$d, so @cycle
my @cycle = ($a,$b,$c,$d);
my $min = minstr(@cycle);
while ($cycle[0] ne $min) { # rotate to $cycle[0] the minimum
push @cycle, (shift @cycle);
}
$callback->(@cycle);
}
}
}
}
return;
}
#------------------------------------------------------------------------------
# Euler Cycle
# Return a list of vertices v1,v2,...,vn,v1 which is an Euler cycle, so
# traverse each edge exactly once.
#
sub Graph_Euler_cycle {
my ($graph, %options) = @_;
my $type = $options{'type'} || 'cycle';
### $type
my @vertices = $graph->vertices;
my $func = cmp_func(@vertices);
@vertices = sort $func @vertices;
my @edges = $graph->edges;
my $num_edges = scalar(@edges);
my @edge_keys = map {join(' to ',@$_)} @edges;
my %edge_keys = map { my $key = join(' to ',@$_);
($key => $key,
join(' to ',reverse @$_) => $key)
} @edges;
my %neighbours;
foreach my $v (@vertices) {
$neighbours{$v} = [ sort $func $graph->neighbours($v) ];
}
my @path = $vertices[0];
my $try;
$try = sub {
my ($visited) = @_;
if (scalar(keys %$visited) >= $num_edges) {
return 1;
}
my $v = $path[-1];
foreach my $to (@{$neighbours{$v}}) {
my $edge = $edge_keys{"$v to $to"};
next if $visited->{$edge};
push @path, $to;
if ($try->({ %$visited, $edge => 1 })) {
return 1;
}
pop @path;
}
return 0;
};
if ($try->({})) {
return @path;
} else {
return;
}
# my @path;
# my %visited;
# my $v = $vertices[0];
# my @nn = (-1);
# my $upto = 0;
# for (;;) {
# my $v = $path[$upto];
# my $n = ++$nn[$upto];
# my $to = $neighbours{$v}->[$n];
# ### at: join('--',@path) . " upto=$upto v=$v n=$n"
# ### $to
# ### assert: 0 <= $n && $n <= $#{$neighbours{$v}}+1
# if (! defined $to) {
# ### no more neighbours, backtrack ...
# $visited{$v} = 0;
# $upto--;
# last if $upto < 0;
# next;
# }
# if ($visited{$to}) {
# ### to is visited ...
# if ($upto == $num_vertices-1
# && ($type eq 'path'
# || $to eq $path[0])) {
# ### found path or cycle ...
# if ($options{'verbose'}) { print "found ",join(',',@path),"\n"; }
# if ($options{'found_coderef'}) { $options{'found_coderef'}->(@path); }
# if (! $options{'all'}) { return 1; }
# }
# next;
# }
#
# # extend path to $to
# $upto++;
# $path[$upto] = $to;
# $visited{$to} = 1;
# $nn[$upto] = -1;
# }
}
#------------------------------------------------------------------------------
# Hamiltonian Cycle
# $graph is a Graph.pm.
# Return true if it has a Hamiltonian cycle (a cycle visiting all vertices
# once each). Key/value options are
#
# type => "cycle" or "path" (default "cycle")
#
# type "path" means search for a Hamiltonian path (a path visiting all
# vertices once each).
#
# Currently this is a depth first search so quite slow and suitable only for
# a small number of vertices.
#
sub Graph_is_Hamiltonian {
my ($graph, %options) = @_;
my $type = $options{'type'} || 'cycle';
### $type
my @vertices = $graph->vertices;
my $num_vertices = scalar(@vertices);
my %neighbours;
foreach my $v (@vertices) {
$neighbours{$v} = [ $graph->neighbours($v) ];
}
foreach my $start (defined $options{'start'} ? $options{'start'}
: $type eq 'path' ? @vertices
: $vertices[0]) {
if ($options{'verbose'}) { print "try start $start\n"; }
my @path = ($start);
my %visited = ($path[0] => 1);
my @nn = (-1);
my $upto = 0;
for (;;) {
my $v = $path[$upto];
my $n = ++$nn[$upto];
my $to = $neighbours{$v}->[$n];
### at: join('--',@path) . " upto=$upto v=$v n=$n"
### $to
### assert: 0 <= $n && $n <= $#{$neighbours{$v}}+1
if (! defined $to) {
### no more neighbours, backtrack ...
$visited{$v} = 0;
$upto--;
last if $upto < 0;
next;
}
if ($visited{$to}) {
### to is visited ...
if ($upto == $num_vertices-1
&& ($type eq 'path'
|| $to eq $path[0])) {
### found path or cycle ...
if ($options{'verbose'}) { print "found ",join(',',@path),"\n"; }
if ($options{'found_coderef'}) { $options{'found_coderef'}->(@path); }
if (! $options{'all'}) { return 1; }
}
next;
}
# extend path to $to
$upto++;
$path[$upto] = $to;
$visited{$to} = 1;
$nn[$upto] = -1;
}
}
return 0;
}
#------------------------------------------------------------------------------
# Directed Graphs
# $graph is a directed Graph.pm.
# Return the number of maximal paths.
# A maximal path is from a predecessorless to a successorless.
# There might be multiple paths between a given predecessorless and
# successorless. All such paths are counted.
#
sub Graph_num_maximal_paths {
my ($graph) = @_;
### Graph_num_maximal_paths() ...
$graph->expect_directed;
my %indegree_remaining;
my %ways;
my %pending;
foreach my $v ($graph->vertices) {
$pending{$v} = 1;
if ($indegree_remaining{$v} = $graph->in_degree($v)) {
$ways{$v} = 0;
} else {
$ways{$v} = 1;
}
}
my $ret = 0;
while (%pending) {
### at pending: scalar(keys %pending)
my $progress;
foreach my $v (keys %pending) {
if ($indegree_remaining{$v}) {
### not ready: "$v indegree_remaining $indegree_remaining{$v}"
### assert: $indegree_remaining{$v} >= 0
next;
}
delete $pending{$v};
my @successors = $graph->successors($v);
if (@successors) {
foreach my $to (@successors) {
### edge: "$v to $to countedge ".$graph->get_edge_count($v,$to)
$pending{$to} or die "oops, to=$to not pending";
$ways{$to} += $ways{$v} * $graph->get_edge_count($v,$to);
$indegree_remaining{$to}--;
$progress = 1;
}
} else {
# successorless
$ret += $ways{$v};
}
}
if (%pending && !$progress) {
die "Graph_num_maximal_paths() oops, no progress, circular graph";
}
}
return $ret;
}
#------------------------------------------------------------------------------
# Lattices
# $graph is a directed Graph.pm.
# Return the number of pairs of comparable elements $u,$v, meaning pairs
# where there is a path from $u to $v. The count includes $u,$u empty path.
# For a lattice graph, this is the number of "intervals" in the lattice.
#
sub Graph_num_intervals {
my ($graph) = @_;
my $ret = 0;
foreach my $v ($graph->vertices) {
$ret += 1 + $graph->all_successors($v);
}
return $ret;
}
sub Graph_successors_matrix {
my ($graph, $vertices_aref, $vertex_to_index_href) = @_;
### $vertices_aref
### $vertex_to_index_href
my @ret;
foreach my $i_from (0 .. $#$vertices_aref) {
foreach my $to ($graph->successors($vertices_aref->[$i_from])) {
my $i_to = $vertex_to_index_href->{$to}
// die "oops, not found: $to";
$ret[$i_from]->[$i_to] = 1;
}
}
return \@ret;
}
sub Graph_reachable_matrix {
my ($graph, $vertices_aref, $vertex_to_index_href) = @_;
my $ret
= Graph_successors_matrix($graph,$vertices_aref,$vertex_to_index_href);
foreach my $i (0 .. $#$vertices_aref) {
$ret->[$i]->[$i] = 1;
}
my $more = 1;
while ($more) {
$more = 0;
foreach my $i (0 .. $#$vertices_aref) {
foreach my $j (0 .. $#$vertices_aref) {
foreach my $k (0 .. $#$vertices_aref) {
if ($ret->[$i]->[$j] && $ret->[$j]->[$k]
&& ! $ret->[$i]->[$k]) {
$ret->[$i]->[$k] = 1;
$more = 1;
}
}
}
}
}
return $ret;
}
# $graph is a directed Graph.pm which is a lattice.
# Return its "intervals lattice".
#
# An interval is a pair [$x,$y] with $y reachable from $x.
# Each vertex of the intervals lattice is such an interval, in the form of a
# string "$x-$y". Edges are from "$x-$y" to "$u-$v" where $x < $u and $y < $v,
# where < means $u reachable from $x, and $v reachable from $y.
#
sub Graph_make_intervals_lattice {
my ($graph, $covers) = @_;
$graph->expect_directed;
my $intervals = Graph->new;
my @vertices = $graph->vertices;
my %vertex_to_index;
@vertex_to_index{@vertices} = (0 .. $#vertices);
my $graph_reachable
= Graph_reachable_matrix($graph, \@vertices, \%vertex_to_index);
### $graph_reachable
sum(map{sum(map {$_||0} @$_)} @$graph_reachable) == Graph_num_intervals($graph) or die;
my %intervals;
foreach my $a (0 .. $#vertices) {
foreach my $b (0 .. $#vertices) {
next unless $graph_reachable->[$a]->[$b];
my $from = "$vertices[$a]-$vertices[$b]";
$intervals->add_vertex($from);
$intervals{$from} = [$a,$b];
}
}
foreach my $from (keys %intervals) {
my $from_aref = $intervals{$from};
foreach my $to (keys %intervals) {
next if $to eq $from;
my $to_aref = $intervals{$to};
next unless $graph_reachable->[$from_aref->[0]]->[$to_aref->[0]];
next unless $graph_reachable->[$from_aref->[1]]->[$to_aref->[1]];
### $from
### $to
# print "$a $b $c $d\n";
# next if $covers && defined $intervals->path_length($from,$to);
$intervals->add_edge($from, $to);
}
}
return $covers ? Graph_covers($intervals) : $intervals;
# $graph->expect_directed;
# my $intervals = Graph->new;
# foreach my $a ($graph->vertices) {
# foreach my $b ($graph->vertices) {
# next unless defined $graph->path_length($a,$b);
# my $from = "$a -- $b";
#
# foreach my $c ($graph->vertices) {
# next unless defined $graph->path_length($a,$c);
# foreach my $d ($graph->vertices) {
# next unless defined $graph->path_length($c,$d);
# next unless defined $graph->path_length($b,$d);
# my $to = "$c -- $d";
# next if $to eq $from;
# # print "$a $b $c $d\n";
# next if $covers && defined $intervals->path_length($from,$to);
# $intervals->add_edge($from, $to);
# }
# }
# }
# }
# return $covers ? Graph_covers($intervals) : $intervals;
}
# $graph is a directed Graph.pm which is expected to be acyclic.
# Delete edges to leave just its cover relations.
#
# At some from->to, if there is also from->mid->to then edge from->to is not
# a cover and is deleted.
#
sub Graph_covers {
my ($graph) = @_;
$graph->expect_acyclic;
my @vertices = $graph->vertices;
my %vertex_to_index;
@vertex_to_index{@vertices} = (0 .. $#vertices);
my $reachable
= Graph_reachable_matrix($graph, \@vertices, \%vertex_to_index);
foreach my $from (0 .. $#vertices) {
foreach my $mid (0 .. $#vertices) {
next if $from == $mid;
next unless $reachable->[$from]->[$mid];
foreach my $to (0 .. $#vertices) {
next if $mid == $to;
next unless $reachable->[$mid]->[$to];
$graph->delete_edge($vertices[$from],$vertices[$to]);
}
}
}
return $graph;
}
# $graph is a directed Graph.pm which is expected to be a lattice.
# Return its unique lowest element.
sub Graph_lattice_lowest {
my ($graph) = @_;
my @predecessorless = $graph->predecessorless_vertices;
@predecessorless==1
or die "Graph_lattice_lowest() oops, expected one predecessorless";
return $predecessorless[0];
}
# $graph is a directed Graph.pm which is expected to be a lattice.
# Return its unique highest element.
sub Graph_lattice_highest {
my ($graph) = @_;
my @successorless = $graph->successorless_vertices;
@successorless==1
or die "Graph_lattice_highest() oops, expected one successorless";
return $successorless[0];
}
# $graph is a directed Graph.pm which is expected to be a lattice.
# Return $href where
# $href->{'max'}->{$x}->{$y} is the lattice max($x,y)
# $href->{'min'}->{$x}->{$y} is the lattice min($x,y)
#
sub Graph_lattice_minmax_hash {
my ($graph) = @_;
my $verbose = 1;
my %hash;
my @vertices = $graph->vertices;
foreach my $elem (['all_successors','max'],
['all_predecessors','min']) {
my ($all_method, $key) = @$elem;
# $all_successors{$x}->{$y} = boolean, true x has y after it, false if not.
# x is a successor of itself ($graph->all_successors doesn't include x
# itself).
my %all_successors;
foreach my $x (@vertices) {
$all_successors{$x}->{$x} = 1;
foreach my $s ($graph->$all_method($x)) {
$all_successors{$x}->{$s} = 1;
}
}
# For each pair x,y look at the common successors and choose the smallest.
# Smallest in the sense the smaller has bigger among its successors.
foreach my $x (@vertices) {
my $xs_href = $all_successors{$x};
foreach my $y (@vertices) {
my $ys_href = $all_successors{$y};
my $m;
foreach my $xs (keys %$xs_href) {
if ($ys_href->{$xs}) { # common successor
if (!defined $m || $all_successors{$xs}->{$m}) {
$m = $xs; # which is before best $m so far
}
}
}
$hash{$key}->{$x}->{$y} = $m;
}
}
}
return \%hash;
# foreach my $v (@vertices) {
# $hash{'max'}->{$v}->{$v}
# = $hash{'min'}->{$v}->{$v} = $v;
# }
# foreach my $x (@vertices) {
# foreach my $y ($graph->all_successors($x)) {
# $hash{'max'}->{$x}->{$y}
# = $hash{'max'}->{$y}->{$x} = $y;
# if ($verbose) { print "successor $x max $y = $y\n"; }
# }
# foreach my $y ($graph->all_predecessors($x)) {
# $hash{'min'}->{$x}->{$y}
# = $hash{'min'}->{$y}->{$x} = $y;
# if ($verbose) { print "predecessor $x min $y = $y\n"; }
# }
# }
# my $more = 1;
# while ($more) {
# $more = 0;
# foreach my $M ('min','max') {
# foreach my $x (@vertices) {
# foreach my $y (@vertices) {
# if (defined(my $m $hash{$M}->{$x}->{$y})) {
# foreach my $z (@vertices) {
#
# if (defined(my $m = $hash{'max'}->{$y}->{$z})) {
# $more = 1;
# $hash{'max'}->{$x}->{$y}
# = $hash{'max'}->{$y}->{$x}
# = $m;
# if ($verbose) { print "chain $x max $y = $m from $z\n"; }
# }
# }
# }
# if (! defined $hash{'min'}->{$x}->{$y}) {
# foreach my $z ($graph->predecessors($y)) {
# if (defined(my $m = $hash{'min'}->{$x}->{$z})) {
# $more = 1;
# $hash{'min'}->{$x}->{$y}
# = $hash{'min'}->{$y}->{$x}
# = $m;
# if ($verbose) { print "chain $x min $y = $m from $z\n"; }
# }
# }
# }
# }
# }
# }
# my $more = 1;
# while ($more) {
# $more = 0;
# foreach my $x (@vertices) {
# foreach my $y (@vertices) {
# if (! defined $hash{'max'}->{$x}->{$y}) {
# foreach my $z ($graph->successors($y)) {
# if (defined(my $m = $hash{'max'}->{$x}->{$z})) {
# $more = 1;
# $hash{'max'}->{$x}->{$y}
# = $hash{'max'}->{$y}->{$x}
# = $m;
# if ($verbose) { print "chain $x max $y = $m from $z\n"; }
# }
# }
# }
# if (! defined $hash{'min'}->{$x}->{$y}) {
# foreach my $z ($graph->predecessors($y)) {
# if (defined(my $m = $hash{'min'}->{$x}->{$z})) {
# $more = 1;
# $hash{'min'}->{$x}->{$y}
# = $hash{'min'}->{$y}->{$x}
# = $m;
# if ($verbose) { print "chain $x min $y = $m from $z\n"; }
# }
# }
# }
# }
# }
# }
#
# return \%hash;
}
# $graph is a directed Graph.pm which is expected to be a lattice.
# $href is a hashref as returned by Graph_lattice_minmax_hash().
# Check that the relations in $href follow the lattice rules.
# die() if bad.
#
sub Graph_lattice_minmax_validate {
my ($graph, $href) = @_;
my $str = Graph_lattice_minmax_reason($graph,$href);
if ($str) {
die 'Graph_lattice_minmax_validate() ', $str;
}
}
# $graph is a directed Graph.pm which is expected to be a lattice.
# $href is a hashref as returned by Graph_lattice_minmax_hash().
# Check that the relations in $href follow the lattice rules.
# If good then return empty string ''.
# If bad then return a string describing the problem.
#
sub Graph_lattice_minmax_reason {
my ($graph, $href) = @_;
# defined
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
foreach my $M ('min','max') {
defined $href->{$M}->{$x}->{$y}
or return "missing $x $M $y";
}
}
}
# commutative
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
foreach my $M ('min','max') {
$href->{$M}->{$x}->{$y} eq $href->{$M}->{$y}->{$x}
or return "not commutative $x $M $y";
}
}
}
# idempotent
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
foreach my $M ('min','max') {
my $m = $href->{$M}->{$x}->{$y};
$href->{$M}->{$x}->{$m} eq $m
or return "not idempotent $x $M $y";
}
}
}
# absorptive a ^ (a v b) = a v (a ^ b) = a
# L H
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
my $min = $href->{'min'}->{$x}->{$y};
my $max = $href->{'max'}->{$x}->{$y};
my $a = $href->{'max'}->{$x}->{$min};
my $b = $href->{'min'}->{$x}->{$max};
($a eq $x && $b eq $x)
or return "not absorptive $x and $y min $min max $max got $a and $b";
}
}
# associative (xy)z = x(yz)
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
foreach my $z ($graph->vertices) {
foreach my $M ('min','max') {
my $a = $href->{$M}->{$href->{$M}->{$x}->{$y}}->{$z};
my $b = $href->{$M}->{$x}->{$href->{$M}->{$y}->{$z}};
$a eq $b
or return "not associative $x $M $y $M $z got $a and $b";
}
}
}
}
return '';
}
# $graph is a directed Graph.pm which is a lattice.
# $href is a hashref as returned by Graph_lattice_minmax_hash().
# Return true if $graph is semi-distributive.
#
sub lattice_minmax_is_semidistributive {
my ($graph, $href) = @_;
foreach my $x ($graph->vertices) {
foreach my $y ($graph->vertices) {
my $m = $href->{'min'}->{$x}->{$y};
my $M = $href->{'max'}->{$x}->{$y};
foreach my $z ($graph->vertices) {
if ($m eq $href->{'min'}->{$x}->{$z}) {
$href->{'min'}->{$x}->{$href->{'max'}->{$y}->{$z}} eq $m
or return 0;
}
if ($M eq $href->{'max'}->{$x}->{$z}) {
$href->{'max'}->{$x}->{$href->{'min'}->{$y}->{$z}} eq $M
or return 0;
}
}
}
}
}
# $graph is a directed Graph.pm which is a lattice.
# $href is a hashref as returned by Graph_lattice_minmax_hash().
# Return the number of complementary pairs in $graph.
# A complementary pair is vertices u,v where
# min(u,v) = global min and max(u,v) = global max
# so they neither meet nor join other than the global min,max.
#
# u = global min and v = global max is always a complementary pair.
# If the lattice is just 1 vertex then this includes u=v as a pair.
#
sub lattice_minmax_num_complementary_pairs {
my ($graph, $href) = @_;
my $lowest = MyGraphs::Graph_lattice_lowest($graph);
my $highest = MyGraphs::Graph_lattice_highest($graph);
my @vertices = $graph->vertices;
my $count_complementary = 0;
foreach my $i (0 .. $#vertices) {
my $u = $vertices[$i];
foreach my $j ($i .. $#vertices) {
my $v = $vertices[$j];
my $min = $href->{'min'}->{$u}->{$v};
my $max = $href->{'max'}->{$u}->{$v};
$count_complementary += ($min eq $lowest && $max eq $highest);
}
}
return $count_complementary;
}
# Think not efficient to check pair-by-pair.
#
# # Return true if $u and $v are complementary, meaning their min is the
# # bottom element and max is the top element.
# sub lattice_is_complementary {
# my ($graph, $u,$v) = @_;
# return lattice_min($graph, $u,$v) eq Graph_lattice_lowest($graph)
# && lattice_max($graph, $u,$v) eq Graph_lattice_highest($graph);
# }
# Is it efficient to search lattice min(x,y) or max(x,y), or better always
# build whole table?
#
# sub lattice_min {
# my ($graph, $u, $v) = @_;
# return lattice_min_or_max($graph,$u,$v, 'predecessors', 'all_predecessors');
# }
# sub lattice_max {
# my ($graph, $u, $v) = @_;
# return lattice_min_or_max($graph,$u,$v, 'successors', 'all_successors');
# }
# sub lattice_min_or_max {
# my ($graph, $u, $v, $immediate, $all) = @_;
#
# die "WRONG";
#
# my @verts = ($u,$v);
# my @verts_descendants;
# foreach my $i (0,1) {
# $verts_descendants[$i]->[0]->{$verts[$i]} = 1;
# }
# for (my $distance = 0; ; $distance++) {
# foreach my $i (0,1) {
# foreach my $from (keys %{$verts_descendants[$i]->[$distance]}) {
# foreach my $to_distance (0 .. $distance) {
# if ($verts_descendants[!$i]->[$to_distance]->{$from}) {
# return $from;
# }
# }
# }
# }
# foreach my $i (0,1) {
# $verts_descendants[$i]->[$distance+1]
# = graph_following_set_hashref($graph,$immediate,
# $verts_descendants[$i]->[$distance]);
# }
# if (! $verts_descendants[0]->[$distance+1]
# && ! $verts_descendants[1]->[$distance+1]) {
# die "lattice_min_or_max() not found";
# }
# }
#
# # my %v_successors; @v_successors{$v, $graph->$all($v)} = (); # hash slice
# # my %t = ($u => 1);
# # while (%t) {
# # foreach my $t (keys %t) {
# # if (exists $v_successors{$t}) {
# # return $t;
# # }
# # }
# # my %new_t;
# # foreach my $t (keys %t) {
# # @new_t{$graph->$immediate($t)} = (); # hash slice
# # }
# # %t = %new_t;
# # }
# # die "lattice_min_or_max() not found";
# }
# sub graph_following_set_hashref {
# my ($graph, $method, $href) = @_;
# my %ret;
# foreach my $v (keys %$href) {
# @ret{$graph->$method($v)} = (); # hash slice
# }
# return \%ret;
# }
#------------------------------------------------------------------------------
1;
__END__
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