Provided by: libgraph-perl_0.9704-1_all bug

NAME

       Graph - graph data structures and algorithms

SYNOPSIS

               use Graph;
               my $g0 = Graph->new;             # A directed graph.

               use Graph::Directed;
               my $g1 = Graph::Directed->new;   # A directed graph.

               use Graph::Undirected;
               my $g2 = Graph::Undirected->new; # An undirected graph.

               $g->add_edge(...);
               $g->has_edge(...)
               $g->delete_edge(...);

               $g->add_vertex(...);
               $g->has_vertex(...);
               $g->delete_vertex(...);

               $g->vertices(...)
               $g->edges(...)

               # And many, many more, see below.

DESCRIPTION

   Non-Description
       This module is not for drawing or rendering any sort of graphics or images, business,
       visualization, or otherwise.

   Description
       Instead, this module is for creating abstract data structures called graphs, and for doing
       various operations on those.

   Perl 5.6.0 minimum
       The implementation depends on a Perl feature called "weak references" and Perl 5.6.0 was
       the first to have those.

   Constructors
       new Create an empty graph.

       Graph->new(%options)
           The options are a hash with option names as the hash keys and the option values as the
           hash values.

           The following options are available:

           directed
                   A boolean option telling that a directed graph should be created.  Often
                   somewhat redundant because a directed graph is the default for the Graph class
                   or one could simply use the "new()" constructor of the Graph::Directed class.

                   You can test the directness of a graph with $g->is_directed() and
                   $g->is_undirected().

           undirected
                   A boolean option telling that an undirected graph should be created.  One
                   could also use the "new()" constructor the Graph::Undirected class instead.

                   Note that while often it is possible to think undirected graphs as
                   bidirectional graphs, or as directed graphs with edges going both ways, in
                   this module directed graphs and undirected graphs are two different things
                   that often behave differently.

                   You can test the directness of a graph with $g->is_directed() and
                   $g->is_undirected().

           refvertexed
           refvertexed_stringified
                   If you want to use references (including Perl objects) as vertices, use
                   "refvertexed".

                   Note that using "refvertexed" means that internally the memory address of the
                   reference (for example, a Perl object) is used as the "identifier" of the
                   vertex, not the stringified form of the reference, even if you have defined
                   your own stringification using "overload".

                   This avoids the problem of the stringified references potentially being
                   identical (because they are identical in value, for example) even if the
                   references are different.  If you really want to use references and their
                   stringified forms as the identities, use the "refvertexed_stringified".  But
                   please do not stringify different objects to the same stringified value.

           unionfind
                   If the graph is undirected, you can specify the "unionfind" parameter to use
                   the so-called union-find scheme to speed up the computation of connected
                   components of the graph (see "is_connected", "connected_components",
                   "connected_component_by_vertex", "connected_component_by_index", and
                   "same_connected_components").  If "unionfind" is used, adding edges (and
                   vertices) becomes slower, but connectedness queries become faster.  You must
                   not delete egdes or vertices of an unionfind graph, only add them.  You can
                   test a graph for "union-findness" with

           has_union_find
                   Returns true if the graph was created with a true "unionfind" parameter.

           vertices
                   An array reference of vertices to add.

           edges   An array reference of array references of edge vertices to add.

       copy
       copy_graph
               my $c = $g->copy_graph;

           Create a shallow copy of the structure (vertices and edges) of the graph.  If you want
           a deep copy that includes attributes, see "deep_copy".  The copy will have the same
           directedness as the original, and if the original was a "compat02" graph, the copy
           will be, too.

           Also the following vertex/edge attributes are copied:

             refvertexed/hypervertexed/countvertexed/multivertexed
             hyperedged/countedged/multiedged/omniedged

           NOTE: You can get an even shallower copy of a graph by

               my $c = $g->new;

           This will copy only the graph properties (directed, and so forth), but none of the
           vertices or edges.

       deep_copy
       deep_copy_graph
               my $c = $g->deep_copy_graph;

           Create a deep copy of the graph (vertices, edges, and attributes) of the graph.  If
           you want a shallow copy that does not include attributes, see "copy".

           Note that copying code references only works with Perls 5.8 or later, and even then
           only if B::Deparse can reconstruct your code.  This functionality uses either Storable
           or Data::Dumper behind the scenes, depending on which is available (Storable is
           preferred).

       undirected_copy
       undirected_copy_graph
               my $c = $g->undirected_copy_graph;

           Create an undirected shallow copy (vertices and edges) of the directed graph so that
           for any directed edge (u, v) there is an undirected edge (u, v).

       undirected_copy_clear_cache
               @path = $g->undirected_copy_clear_cache;

           See "Clearing cached results".

       directed_copy
       directed_copy_graph
               my $c = $g->directed_copy_graph;

           Create a directed shallow copy (vertices and edges) of the undirected graph so that
           for any undirected edge (u, v) there are two directed edges (u, v) and (v, u).

       transpose
       transpose_graph
               my $t = $g->transpose_graph;

           Create a directed shallow transposed copy (vertices and edges) of the directed graph
           so that for any directed edge (u, v) there is a directed edge (v, u).

           You can also transpose a single edge with

           transpose_edge
                       $g->transpose_edge($u, $v)

       complete_graph
       complete
               my $c = $g->complete_graph;

           Create a complete graph that has the same vertices as the original graph.  A complete
           graph has an edge between every pair of vertices.

       complement_graph
       complement
               my $c = $g->complement_graph;

           Create a complement graph that has the same vertices as the original graph.  A
           complement graph has an edge (u,v) if and only if the original graph does not have
           edge (u,v).

       subgraph
              my $c = $g->subgraph(\@src, \@dst);
              my $c = $g->subgraph(\@src);

           Creates a subgraph of a given graph.  The created subgraph has the same graph
           properties (directedness, and so forth) as the original graph, but none of the
           attributes (graph, vertex, or edge).

           A vertex is added to the subgraph if it is in the original graph.

           An edge is added to the subgraph if there is an edge in the original graph that starts
           from the "src" set of vertices and ends in the "dst" set of vertices.

           You can leave out "dst" in which case "dst" is assumed to be the same: this is called
           a vertex-induced subgraph.

       See also "random_graph" for a random constructor.

   Basics
       add_vertex
               $g->add_vertex($v)

           Add the vertex to the graph.  Returns the graph.

           By default idempotent, but a graph can be created countvertexed.

           A vertex is also known as a node.

           Adding "undef" as vertex is not allowed.

           Note that unless you have isolated vertices (or countvertexed vertices), you do not
           need to explicitly use "add_vertex" since "add_edge" will implicitly add its vertices.

       add_edge
               $g->add_edge($u, $v)

           Add the edge to the graph.  Implicitly first adds the vertices if the graph does not
           have them.  Returns the graph.

           By default idempotent, but a graph can be created countedged.

           An edge is also known as an arc.

       has_vertex
               $g->has_vertex($v)

           Return true if the vertex exists in the graph, false otherwise.

       has_edge
               $g->has_edge($u, $v)

           Return true if the edge exists in the graph, false otherwise.

       delete_vertex
               $g->delete_vertex($v)

           Delete the vertex from the graph.  Returns the graph, even if the vertex did not exist
           in the graph.

           If the graph has been created multivertexed or countvertexed and a vertex has been
           added multiple times, the vertex will require at least an equal number of deletions to
           become completely deleted.

       delete_vertices
               $g->delete_vertices($v1, $v2, ...)

           Delete the vertices from the graph.  Returns the graph, even if none of the vertices
           existed in the graph.

           If the graph has been created multivertexed or countvertexed and a vertex has been
           added multiple times, the vertex will require at least an equal number of deletions to
           become completely deleteted.

       delete_edge
               $g->delete_edge($u, $v)

           Delete the edge from the graph.  Returns the graph, even if the edge did not exist in
           the graph.

           If the graph has been created multivertexed or countedged and an edge has been added
           multiple times, the edge will require at least an equal number of deletions to become
           completely deleted.

       delete_edges
               $g->delete_edges($u1, $v1, $u2, $v2, ...)

           Delete the edges from the graph.  Returns the graph, even if none of the edges existed
           in the graph.

           If the graph has been created multivertexed or countedged and an edge has been added
           multiple times, the edge will require at least an equal number of deletions to become
           completely deleted.

   Displaying
       Graphs have stringification overload, so you can do things like

           print "The graph is $g\n"

       One-way (directed, unidirected) edges are shown as '-', two-way (undirected, bidirected)
       edges are shown as '='.  If you want to, you can call the stringification via the method

       stringify

   Boolean
       Graphs have boolifying overload, so you can do things like

           if ($g) { print "The graph is: $g\n" }

       which works even if the graph is empty.  In fact, the boolify always returns true.  If you
       want to test for example for vertices, test for vertices.

       boolify

   Comparing
       Testing for equality can be done either by the overloaded "eq" operator

           $g eq "a-b,a-c,d"

       or by the method

       eq
               $g->eq("a-b,a-c,d")

       The equality testing compares the stringified forms, and therefore it assumes total
       equality, not isomorphism: all the vertices must be named the same, and they must have
       identical edges between them.

       For unequality there are correspondingly the overloaded "ne" operator and the method

       ne
               $g->ne("a-b,a-c,d")

       See also "Isomorphism".

   Paths and Cycles
       Paths and cycles are simple extensions of edges: paths are edges starting from where the
       previous edge ended, and cycles are paths returning back to the start vertex of the first
       edge.

       add_path
              $g->add_path($a, $b, $c, ..., $x, $y, $z)

           Add the edges $a-$b, $b-$c, ..., $x-$y, $y-$z to the graph.  Returns the graph.

       has_path
              $g->has_path($a, $b, $c, ..., $x, $y, $z)

           Return true if the graph has all the edges $a-$b, $b-$c, ..., $x-$y, $y-$z, false
           otherwise.

       delete_path
              $g->delete_path($a, $b, $c, ..., $x, $y, $z)

           Delete all the edges edges $a-$b, $b-$c, ..., $x-$y, $y-$z (regardless of whether they
           exist or not).  Returns the graph.

       add_cycle
              $g->add_cycle($a, $b, $c, ..., $x, $y, $z)

           Add the edges $a-$b, $b-$c, ..., $x-$y, $y-$z, and $z-$a to the graph.  Returns the
           graph.

       has_cycle
       has_this_cycle
              $g->has_cycle($a, $b, $c, ..., $x, $y, $z)

           Return true if the graph has all the edges $a-$b, $b-$c, ..., $x-$y, $y-$z, and $z-$a,
           false otherwise.

           NOTE: This does not detect cycles, see "has_a_cycle" and "find_a_cycle".

       delete_cycle
              $g->delete_cycle($a, $b, $c, ..., $x, $y, $z)

           Delete all the edges edges $a-$b, $b-$c, ..., $x-$y, $y-$z, and $z-$a (regardless of
           whether they exist or not).  Returns the graph.

       has_a_cycle
              $g->has_a_cycle

           Returns true if the graph has a cycle, false if not.

       find_a_cycle
              $g->find_a_cycle

           Returns a cycle if the graph has one (as a list of vertices), an empty list if no
           cycle can be found.

           Note that this just returns the vertices of a cycle: not any particular cycle, just
           the first one it finds.  A repeated call might find the same cycle, or it might find a
           different one, and you cannot call this repeatedly to find all the cycles.

   Graph Types
       is_simple_graph
               $g->is_simple_graph

           Return true if the graph has no multiedges, false otherwise.

       is_pseudo_graph
               $g->is_pseudo_graph

           Return true if the graph has any multiedges or any self-loops, false otherwise.

       is_multi_graph
               $g->is_multi_graph

           Return true if the graph has any multiedges but no self-loops, false otherwise.

       is_directed_acyclic_graph
       is_dag
               $g->is_directed_acyclic_graph
               $g->is_dag

           Return true if the graph is directed and acyclic, false otherwise.

       is_cyclic
               $g->is_cyclic

           Return true if the graph is cyclic (contains at least one cycle).  (This is identical
           to "has_a_cycle".)

           To find at least one such cycle, see "find_a_cycle".

       is_acyclic
           Return true if the graph is acyclic (does not contain any cycles).

       To find a cycle, use "find_a_cycle".

   Transitivity
       is_transitive
               $g->is_transitive

           Return true if the graph is transitive, false otherwise.

       TransitiveClosure_Floyd_Warshall
       transitive_closure
               $tcg = $g->TransitiveClosure_Floyd_Warshall

           Return the transitive closure graph of the graph.

       You can query the reachability from $u to $v with

       is_reachable
               $tcg->is_reachable($u, $v)

       See Graph::TransitiveClosure for more information about creating and querying transitive
       closures.

       With

       transitive_closure_matrix
              $tcm = $g->transitive_closure_matrix;

       you can (create if not existing and) query the transitive closure matrix that underlies
       the transitive closure graph.  See Graph::TransitiveClosure::Matrix for more information.

   Mutators
       add_vertices
               $g->add_vertices('d', 'e', 'f')

           Add zero or more vertices to the graph.  Returns the graph.

       add_edges
               $g->add_edges(['d', 'e'], ['f', 'g'])
               $g->add_edges(qw(d e f g));

           Add zero or more edges to the graph.  The edges are specified as a list of array
           references, or as a list of vertices where the even (0th, 2nd, 4th, ...) items are
           start vertices and the odd (1st, 3rd, 5th, ...) are the corresponding end vertices.
           Returns the graph.

   Accessors
       is_directed
       directed
               $g->is_directed()
               $g->directed()

           Return true if the graph is directed, false otherwise.

       is_undirected
       undirected
               $g->is_undirected()
               $g->undirected()

           Return true if the graph is undirected, false otherwise.

       is_refvertexed
       is_refvertexed_stringified
       refvertexed
       refvertexed_stringified
           Return true if the graph can handle references (including Perl objects) as vertices.

       vertices
               my $V = $g->vertices
               my @V = $g->vertices

           In scalar context, return the number of vertices in the graph.  In list context,
           return the vertices, in no particular order.

       has_vertices
               $g->has_vertices()

           Return true if the graph has any vertices, false otherwise.

       edges
               my $E = $g->edges
               my @E = $g->edges

           In scalar context, return the number of edges in the graph.  In list context, return
           the edges, in no particular order.  The edges are returned as anonymous arrays listing
           the vertices.

       has_edges
               $g->has_edges()

           Return true if the graph has any edges, false otherwise.

       is_connected
               $g->is_connected

           For an undirected graph, return true is the graph is connected, false otherwise.
           Being connected means that from every vertex it is possible to reach every other
           vertex.

           If the graph has been created with a true "unionfind" parameter, the time complexity
           is (essentially) O(V), otherwise O(V log V).

           See also "connected_components", "connected_component_by_index",
           "connected_component_by_vertex", and "same_connected_components", and
           "biconnectivity".

           For directed graphs, see "is_strongly_connected" and "is_weakly_connected".

       connected_components
               @cc = $g->connected_components()

           For an undirected graph, returns the vertices of the connected components of the graph
           as a list of anonymous arrays.  The ordering of the anonymous arrays or the ordering
           of the vertices inside the anonymous arrays (the components) is undefined.

           For directed graphs, see "strongly_connected_components" and
           "weakly_connected_components".

       connected_component_by_vertex
               $i = $g->connected_component_by_vertex($v)

           For an undirected graph, return an index identifying the connected component the
           vertex belongs to, the indexing starting from zero.

           For the inverse, see "connected_component_by_index".

           If the graph has been created with a true "unionfind" parameter, the time complexity
           is (essentially) O(1), otherwise O(V log V).

           See also "biconnectivity".

           For directed graphs, see "strongly_connected_component_by_vertex" and
           "weakly_connected_component_by_vertex".

       connected_component_by_index
               @v = $g->connected_component_by_index($i)

           For an undirected graph, return the vertices of the ith connected component, the
           indexing starting from zero.  The order of vertices is undefined, while the order of
           the connected components is same as from connected_components().

           For the inverse, see "connected_component_by_vertex".

           For directed graphs, see "strongly_connected_component_by_index" and
           "weakly_connected_component_by_index".

       same_connected_components
               $g->same_connected_components($u, $v, ...)

           For an undirected graph, return true if the vertices are in the same connected
           component.

           If the graph has been created with a true "unionfind" parameter, the time complexity
           is (essentially) O(1), otherwise O(V log V).

           For directed graphs, see "same_strongly_connected_components" and
           "same_weakly_connected_components".

       connected_graph
               $cg = $g->connected_graph

           For an undirected graph, return its connected graph.

       connectivity_clear_cache
               $g->connectivity_clear_cache

           See "Clearing cached results".

           See "Connected Graphs and Their Components" for further discussion.

       biconnectivity
               my ($ap, $bc, $br) = $g->biconnectivity

           For an undirected graph, return the various biconnectivity components of the graph:
           the articulation points (cut vertices), biconnected components, and bridges.

           Note: currently only handles connected graphs.

       is_biconnected
              $g->is_biconnected

           For an undirected graph, return true if the graph is biconnected (if it has no
           articulation points, also known as cut vertices).

       is_edge_connected
              $g->is_edge_connected

           For an undirected graph, return true if the graph is edge-connected (if it has no
           bridges).

           Note: more precisely, this would be called is_edge_biconnected, since there is a more
           general concept of being k-connected.

       is_edge_separable
              $g->is_edge_separable

           For an undirected graph, return true if the graph is edge-separable (if it has
           bridges).

           Note: more precisely, this would be called is_edge_biseparable, since there is a more
           general concept of being k-connected.

       articulation_points
       cut_vertices
              $g->articulation_points

           For an undirected graph, return the articulation points (cut vertices) of the graph as
           a list of vertices.  The order is undefined.

       biconnected_components
              $g->biconnected_components

           For an undirected graph, return the biconnected components of the graph as a list of
           anonymous arrays of vertices in the components.  The ordering of the anonymous arrays
           or the ordering of the vertices inside the anonymous arrays (the components) is
           undefined.  Also note that one vertex can belong to more than one biconnected
           component.

       biconnected_component_by_vertex
              $i = $g->biconnected_component_by_index($v)

           For an undirected graph, return the indices identifying the biconnected components the
           vertex belongs to, the indexing starting from zero.  The order of of the components is
           undefined.

           For the inverse, see "connected_component_by_index".

           For directed graphs, see "strongly_connected_component_by_index" and
           "weakly_connected_component_by_index".

       biconnected_component_by_index
              @v = $g->biconnected_component_by_index($i)

           For an undirected graph, return the vertices in the ith biconnected component of the
           graph as an anonymous arrays of vertices in the component.  The ordering of the
           vertices within a component is undefined.  Also note that one vertex can belong to
           more than one biconnected component.

       same_biconnected_components
               $g->same_biconnected_components($u, $v, ...)

           For an undirected graph, return true if the vertices are in the same biconnected
           component.

       biconnected_graph
               $bcg = $g->biconnected_graph

           For an undirected graph, return its biconnected graph.

           See "Connected Graphs and Their Components" for further discussion.

       bridges
              $g->bridges

           For an undirected graph, return the bridges of the graph as a list of anonymous arrays
           of vertices in the bridges.  The order of bridges and the order of vertices in them is
           undefined.

       biconnectivity_clear_cache
               $g->biconnectivity_clear_cache

           See "Clearing cached results".

       strongly_connected
       is_strongly_connected
               $g->is_strongly_connected

           For a directed graph, return true is the directed graph is strongly connected, false
           if not.

           See also "is_weakly_connected".

           For undirected graphs, see "is_connected", or "is_biconnected".

       strongly_connected_component_by_vertex
               $i = $g->strongly_connected_component_by_vertex($v)

           For a directed graph, return an index identifying the strongly connected component the
           vertex belongs to, the indexing starting from zero.

           For the inverse, see "strongly_connected_component_by_index".

           See also "weakly_connected_component_by_vertex".

           For undirected graphs, see "connected_components" or "biconnected_components".

       strongly_connected_component_by_index
               @v = $g->strongly_connected_component_by_index($i)

           For a directed graph, return the vertices of the ith connected component, the indexing
           starting from zero.  The order of vertices within a component is undefined, while the
           order of the connected components is the as from strongly_connected_components().

           For the inverse, see "strongly_connected_component_by_vertex".

           For undirected graphs, see "weakly_connected_component_by_index".

       same_strongly_connected_components
               $g->same_strongly_connected_components($u, $v, ...)

           For a directed graph, return true if the vertices are in the same strongly connected
           component.

           See also "same_weakly_connected_components".

           For undirected graphs, see "same_connected_components" or
           "same_biconnected_components".

       strong_connectivity_clear_cache
               $g->strong_connectivity_clear_cache

           See "Clearing cached results".

       weakly_connected
       is_weakly_connected
               $g->is_weakly_connected

           For a directed graph, return true is the directed graph is weakly connected, false if
           not.

           Weakly connected graph is also known as semiconnected graph.

           See also "is_strongly_connected".

           For undirected graphs, see "is_connected" or "is_biconnected".

       weakly_connected_components
               @wcc = $g->weakly_connected_components()

           For a directed graph, returns the vertices of the weakly connected components of the
           graph as a list of anonymous arrays.  The ordering of the anonymous arrays or the
           ordering of the vertices inside the anonymous arrays (the components) is undefined.

           See also "strongly_connected_components".

           For undirected graphs, see "connected_components" or "biconnected_components".

       weakly_connected_component_by_vertex
               $i = $g->weakly_connected_component_by_vertex($v)

           For a directed graph, return an index identifying the weakly connected component the
           vertex belongs to, the indexing starting from zero.

           For the inverse, see "weakly_connected_component_by_index".

           For undirected graphs, see "connected_component_by_vertex" and
           "biconnected_component_by_vertex".

       weakly_connected_component_by_index
               @v = $g->weakly_connected_component_by_index($i)

           For a directed graph, return the vertices of the ith weakly connected component, the
           indexing starting zero.  The order of vertices within a component is undefined, while
           the order of the weakly connected components is same as from
           weakly_connected_components().

           For the inverse, see "weakly_connected_component_by_vertex".

           For undirected graphs, see connected_component_by_index and
           biconnected_component_by_index.

       same_weakly_connected_components
               $g->same_weakly_connected_components($u, $v, ...)

           Return true if the vertices are in the same weakly connected component.

       weakly_connected_graph
               $wcg = $g->weakly_connected_graph

           For a directed graph, return its weakly connected graph.

           For undirected graphs, see "connected_graph" and "biconnected_graph".

       strongly_connected_components
              my @scc = $g->strongly_connected_components;

           For a directed graph, return the strongly connected components as a list of anonymous
           arrays.  The elements in the anonymous arrays are the vertices belonging to the
           strongly connected component; both the elements and the components are in no
           particular order.

           Note that strongly connected components can have single-element components even
           without self-loops: if a vertex is any of isolated, sink, or a source, the vertex is
           alone in its own strong component.

           See also "weakly_connected_components".

           For undirected graphs, see "connected_components", or see "biconnected_components".

       strongly_connected_graph
              my $scg = $g->strongly_connected_graph;

           See "Connected Graphs and Their Components" for further discussion.

           Strongly connected graphs are also known as kernel graphs.

           See also "weakly_connected_graph".

           For undirected graphs, see "connected_graph", or "biconnected_graph".

       is_sink_vertex
               $g->is_sink_vertex($v)

           Return true if the vertex $v is a sink vertex, false if not.  A sink vertex is defined
           as a vertex with predecessors but no successors: this definition means that isolated
           vertices are not sink vertices.  If you want also isolated vertices, use
           is_successorless_vertex().

       is_source_vertex
               $g->is_source_vertex($v)

           Return true if the vertex $v is a source vertex, false if not.  A source vertex is
           defined as a vertex with successors but no predecessors: the definition means that
           isolated vertices are not source vertices.  If you want also isolated vertices, use
           is_predecessorless_vertex().

       is_successorless_vertex
               $g->is_successorless_vertex($v)

           Return true if the vertex $v has no succcessors (no edges leaving the vertex), false
           if it has.

           Isolated vertices will return true: if you do not want this, use is_sink_vertex().

       is_successorful_vertex
               $g->is_successorful_vertex($v)

           Return true if the vertex $v has successors, false if not.

       is_predecessorless_vertex
               $g->is_predecessorless_vertex($v)

           Return true if the vertex $v has no predecessors (no edges entering the vertex), false
           if it has.

           Isolated vertices will return true: if you do not want this, use is_source_vertex().

       is_predecessorful_vertex
               $g->is_predecessorful_vertex($v)

           Return true if the vertex $v has predecessors, false if not.

       is_isolated_vertex
               $g->is_isolated_vertex($v)

           Return true if the vertex $v is an isolated vertex: no successors and no predecessors.

       is_interior_vertex
               $g->is_interior_vertex($v)

           Return true if the vertex $v is an interior vertex: both successors and predecessors.

       is_exterior_vertex
               $g->is_exterior_vertex($v)

           Return true if the vertex $v is an exterior vertex: has either no successors or no
           predecessors, or neither.

       is_self_loop_vertex
               $g->is_self_loop_vertex($v)

           Return true if the vertex $v is a self loop vertex: has an edge from itself to itself.

       sink_vertices
               @v = $g->sink_vertices()

           Return the sink vertices of the graph.  In scalar context return the number of sink
           vertices.  See "is_sink_vertex" for the definition of a sink vertex.

       source_vertices
               @v = $g->source_vertices()

           Return the source vertices of the graph.  In scalar context return the number of
           source vertices.  See "is_source_vertex" for the definition of a source vertex.

       successorful_vertices
               @v = $g->successorful_vertices()

           Return the successorful vertices of the graph.  In scalar context return the number of
           successorful vertices.

       successorless_vertices
               @v = $g->successorless_vertices()

           Return the successorless vertices of the graph.  In scalar context return the number
           of successorless vertices.

       successors
               @s = $g->successors($v)

           Return the immediate successor vertices of the vertex.

           See also "all_successors", "all_neighbours", and "all_reachable".

       all_successors
               @s = $g->all_successors(@v)

           For a directed graph, returns all successor vertices of the argument vertices,
           recursively.

           For undirected graphs, see "all_neighbours" and "all_reachable".

           See also "successors".

       neighbors
       neighbours
               @n = $g->neighbours($v)

           Return the neighboring/neighbouring vertices.  Also known as the adjacent vertices.

           See also "all_neighbours" and "all_reachable".

       all_neighbors
       all_neighbours
              @n = $g->all_neighbours(@v)

           Return the neighboring/neighbouring vertices of the argument vertices, recursively.
           For a directed graph, recurses up predecessors and down successors.  For an undirected
           graph, returns all the vertices reachable from the argument vertices: equivalent to
           "all_reachable".

           See also "neighbours" and "all_reachable".

       all_reachable
               @r = $g->all_reachable(@v)

           Return all the vertices reachable from of the argument vertices, recursively.  For a
           directed graph, equivalent to "all_successors".  For an undirected graph, equivalent
           to "all_neighbours".  The argument vertices are not included in the results unless
           there are explicit self-loops.

           See also "neighbours", "all_neighbours", and "all_successors".

       predecessorful_vertices
               @v = $g->predecessorful_vertices()

           Return the predecessorful vertices of the graph.  In scalar context return the number
           of predecessorful vertices.

       predecessorless_vertices
               @v = $g->predecessorless_vertices()

           Return the predecessorless vertices of the graph.  In scalar context return the number
           of predecessorless vertices.

       predecessors
               @p = $g->predecessors($v)

           Return the immediate predecessor vertices of the vertex.

           See also "all_predecessors", "all_neighbours", and "all_reachable".

       all_predecessors
               @p = $g->all_predecessors(@v)

           For a directed graph, returns all predecessor vertices of the argument vertices,
           recursively.

           For undirected graphs, see "all_neighbours" and "all_reachable".

           See also "predecessors".

       isolated_vertices
               @v = $g->isolated_vertices()

           Return the isolated vertices of the graph.  In scalar context return the number of
           isolated vertices.  See "is_isolated_vertex" for the definition of an isolated vertex.

       interior_vertices
               @v = $g->interior_vertices()

           Return the interior vertices of the graph.  In scalar context return the number of
           interior vertices.  See "is_interior_vertex" for the definition of an interior vertex.

       exterior_vertices
               @v = $g->exterior_vertices()

           Return the exterior vertices of the graph.  In scalar context return the number of
           exterior vertices.  See "is_exterior_vertex" for the definition of an exterior vertex.

       self_loop_vertices
               @v = $g->self_loop_vertices()

           Return the self-loop vertices of the graph.  In scalar context return the number of
           self-loop vertices.  See "is_self_loop_vertex" for the definition of a self-loop
           vertex.

   Connected Graphs and Their Components
       In this discussion connected graph refers to any of connected graphs, biconnected graphs,
       and strongly connected graphs.

       NOTE: if the vertices of the original graph are Perl objects, (in other words, references,
       so you must be using "refvertexed") the vertices of the connected graph are NOT by default
       usable as Perl objects because they are blessed into a package with a rather unusable
       name.

       By default, the vertex names of the connected graph are formed from the names of the
       vertices of the original graph by (alphabetically sorting them and) concatenating their
       names with "+".  The vertex attribute "subvertices" is also used to store the list (as an
       array reference) of the original vertices.  To change the 'supercomponent' vertex names
       and the whole logic of forming these supercomponents use the "super_component") option to
       the method calls:

         $g->connected_graph(super_component => sub { ... })
         $g->biconnected_graph(super_component => sub { ... })
         $g->strongly_connected_graph(super_component => sub { ... })

       The subroutine reference gets the 'subcomponents' (the vertices of the original graph) as
       arguments, and it is supposed to return the new supercomponent vertex, the "stringified"
       form of which is used as the vertex name.

   Degree
       A vertex has a degree based on the number of incoming and outgoing edges.  This really
       makes sense only for directed graphs.

       degree
       vertex_degree
               $d = $g->degree($v)
               $d = $g->vertex_degree($v)

           For directed graphs: the in-degree minus the out-degree at the vertex.

           For undirected graphs: the number of edges at the vertex  (identical to "in_degree()",
           "out_degree()").

       in_degree
               $d = $g->in_degree($v)

           For directed graphs: the number of incoming edges at the vertex.

           For undirected graphs: the number of edges at the vertex (identical to "out_degree()",
           "degree()", "vertex_degree()").

       out_degree
               $o = $g->out_degree($v)

           For directed graphs: The number of outgoing edges at the vertex.

           For undirected graphs: the number of edges at the vertex (identical to "in_degree()",
           "degree()", "vertex_degree()").

       average_degree
              my $ad = $g->average_degree;

           Return the average degree (as in "degree()" or "vertex_degree()") taken over all
           vertices.

       Related methods are

       edges_at
               @e = $g->edges_at($v)

           The union of edges from and edges to at the vertex.

       edges_from
               @e = $g->edges_from($v)

           The edges leaving the vertex.

       edges_to
               @e = $g->edges_to($v)

           The edges entering the vertex.

       See also "average_degree".

   Counted Vertices
       Counted vertices are vertices with more than one instance, normally adding vertices is
       idempotent.  To enable counted vertices on a graph, give the "countvertexed" parameter a
       true value

           use Graph;
           my $g = Graph->new(countvertexed => 1);

       To find out how many times the vertex has been added:

       get_vertex_count
               my $c = $g->get_vertex_count($v);

           Return the count of the vertex, or undef if the vertex does not exist.

   Multiedges, Multivertices, Multigraphs
       Multiedges are edges with more than one "life", meaning that one has to delete them as
       many times as they have been added.  Normally adding edges is idempotent (in other words,
       adding edges more than once makes no difference).

       There are two kinds or degrees of creating multiedges and multivertices.  The two kinds
       are mutually exclusive.

       The weaker kind is called counted, in which the edge or vertex has a count on it: add
       operations increase the count, and delete operations decrease the count, and once the
       count goes to zero, the edge or vertex is deleted.  If there are attributes, they all are
       attached to the same vertex.  You can think of this as the graph elements being
       refcounted, or reference counted, if that sounds more familiar.

       The stronger kind is called (true) multi, in which the edge or vertex really has multiple
       separate identities, so that you can for example attach different attributes to different
       instances.

       To enable multiedges on a graph:

           use Graph;
           my $g0 = Graph->new(countedged => 1);
           my $g0 = Graph->new(multiedged => 1);

       Similarly for vertices

           use Graph;
           my $g1 = Graph->new(countvertexed => 1);
           my $g1 = Graph->new(multivertexed => 1);

       You can test for these by

       is_countedged
       countedged
               $g->is_countedged
               $g->countedged

           Return true if the graph is countedged.

       is_countvertexed
       countvertexed
               $g->is_countvertexed
               $g->countvertexed

           Return true if the graph is countvertexed.

       is_multiedged
       multiedged
               $g->is_multiedged
               $g->multiedged

           Return true if the graph is multiedged.

       is_multivertexed
       multivertexed
               $g->is_multivertexed
               $g->multivertexed

           Return true if the graph is multivertexed.

       A multiedged (either the weak kind or the strong kind) graph is a multigraph, for which
       you can test with "is_multi_graph()".

       NOTE: The various graph algorithms do not in general work well with multigraphs (they
       often assume simple graphs, that is, no multiedges or loops), and no effort has been made
       to test the algorithms with multigraphs.

       vertices() and edges() will return the multiple elements: if you want just the unique
       elements, use

       unique_vertices
       unique_edges
               @uv = $g->unique_vertices; # unique
               @mv = $g->vertices;        # possible multiples
               @ue = $g->unique_edges;
               @me = $g->edges;

       If you are using (the stronger kind of) multielements, you should use the by_id variants:

       add_vertex_by_id
       has_vertex_by_id
       delete_vertex_by_id
       add_edge_by_id
       has_edge_by_id
       delete_edge_by_id

           $g->add_vertex_by_id($v, $id)
           $g->has_vertex_by_id($v, $id)
           $g->delete_vertex_by_id($v, $id)

           $g->add_edge_by_id($u, $v, $id)
           $g->has_edge_by_id($u, $v, $id)
           $g->delete_edge_by_id($u, $v, $id)

       These interfaces only apply to multivertices and multiedges.  When you delete the last
       vertex/edge in a multivertex/edge, the whole vertex/edge is deleted.  You can use
       add_vertex()/add_edge() on a multivertex/multiedge graph, in which case an id is generated
       automatically.  To find out which the generated id was, you need to use

       add_vertex_get_id
       add_edge_get_id

           $idv = $g->add_vertex_get_id($v)
           $ide = $g->add_edge_get_id($u, $v)

       To return all the ids of vertices/edges in a multivertex/multiedge, use

       get_multivertex_ids
       get_multiedge_ids

           $g->get_multivertex_ids($v)
           $g->get_multiedge_ids($u, $v)

       The ids are returned in random order.

       To find out how many times the edge has been added (this works for either kind of
       multiedges):

       get_edge_count
               my $c = $g->get_edge_count($u, $v);

           Return the count (the "countedness") of the edge, or undef if the edge does not exist.

       The following multi-entity utility functions exist, mirroring the non-multi vertices and
       edges:

       add_weighted_edge_by_id
       add_weighted_edges_by_id
       add_weighted_path_by_id
       add_weighted_vertex_by_id
       add_weighted_vertices_by_id
       delete_edge_weight_by_id
       delete_vertex_weight_by_id
       get_edge_weight_by_id
       get_vertex_weight_by_id
       has_edge_weight_by_id
       has_vertex_weight_by_id
       set_edge_weight_by_id
       set_vertex_weight_by_id

   Topological Sort
       topological_sort
       toposort
               my @ts = $g->topological_sort;

           Return the vertices of the graph sorted topologically.  Note that there may be several
           possible topological orderings; one of them is returned.

           If the graph contains a cycle, a fatal error is thrown, you can either use "eval" to
           trap that, or supply the "empty_if_cyclic" argument with a true value

               my @ts = $g->topological_sort(empty_if_cyclic => 1);

           in which case an empty array is returned if the graph is cyclic.

   Minimum Spanning Trees (MST)
       Minimum Spanning Trees or MSTs are tree subgraphs derived from an undirected graph.  MSTs
       "span the graph" (covering all the vertices) using as lightly weighted (hence the
       "minimum") edges as possible.

       MST_Kruskal
               $mstg = $g->MST_Kruskal;

           Returns the Kruskal MST of the graph.

       MST_Prim
               $mstg = $g->MST_Prim(%opt);

           Returns the Prim MST of the graph.

           You can choose the first vertex with $opt{ first_root }.

       MST_Dijkstra
       minimum_spanning_tree
               $mstg = $g->MST_Dijkstra;
               $mstg = $g->minimum_spanning_tree;

           Aliases for MST_Prim.

   Single-Source Shortest Paths (SSSP)
       Single-source shortest paths, also known as Shortest Path Trees (SPTs).  For either a
       directed or an undirected graph, return a (tree) subgraph that from a single start vertex
       (the "single source") travels the shortest possible paths (the paths with the lightest
       weights) to all the other vertices.  Note that the SSSP is neither reflexive (the shortest
       paths do not include the zero-length path from the source vertex to the source vertex) nor
       transitive (the shortest paths do not include transitive closure paths).  If no weight is
       defined for an edge, 1 (one) is assumed.

       SPT_Dijkstra
               $sptg = $g->SPT_Dijkstra($root)
               $sptg = $g->SPT_Dijkstra(%opt)

           Return as a graph the the single-source shortest paths of the graph using Dijkstra's
           algorithm.  The graph cannot contain negative edges (negative edges cause the
           algorithm to abort with an error message "Graph::SPT_Dijkstra: edge ... is negative").

           You can choose the first vertex of the result with either a single vertex argument or
           with $opt{ first_root }, otherwise a random vertex is chosen.

           NOTE: note that all the vertices might not be reachable from the selected (explicit or
           random) start vertex.

           NOTE: after the first reachable tree from the first start vertex has been finished,
           and if there still are unvisited vertices, SPT_Dijkstra will keep on selecting
           unvisited vertices.

           The next roots (in case the first tree doesn't visit all the vertices) can be chosen
           by setting one of the following options to true: "next_root", "next_alphabetic",
           "next_numeric", "next_random".

           The "next_root" is the most customizable: the value needs to be a subroutine reference
           which will receive the graph and the unvisited vertices as hash reference.  If you
           want to only visit the first tree, use "next_root =" sub { undef }>.  The rest of
           these options are booleans.  If none of them are true, a random unvisited vertex will
           be selected.

           The first start vertex is be available as the graph attribute "SPT_Dijkstra_root").

           The result weights of vertices can be retrieved from the result graph by

                   my $w = $sptg->get_vertex_attribute($v, 'weight');

           The predecessor vertex of a vertex in the result graph can be retrieved by

                   my $u = $sptg->get_vertex_attribute($v, 'p');

           ("A successor vertex" cannot be retrieved as simply because a single vertex can have
           several successors.  You can first find the "neighbors()" vertices and then remove the
           predecessor vertex.)

           If you want to find the shortest path between two vertices, see "SP_Dijkstra".

       SSSP_Dijkstra
       single_source_shortest_paths
           Aliases for SPT_Dijkstra.

       SP_Dijkstra
               @path = $g->SP_Dijkstra($u, $v)

           Return the vertices in the shortest path in the graph $g between the two vertices $u,
           $v.  If no path can be found, an empty list is returned.

           Uses SPT_Dijkstra().

       SPT_Dijkstra_clear_cache
               $g->SPT_Dijkstra_clear_cache

           See "Clearing cached results".

       SPT_Bellman_Ford
               $sptg = $g->SPT_Bellman_Ford(%opt)

           Return as a graph the single-source shortest paths of the graph using Bellman-Ford's
           algorithm.  The graph can contain negative edges but not negative cycles (negative
           cycles cause the algorithm to abort with an error message "Graph::SPT_Bellman_Ford:
           negative cycle exists/").

           You can choose the start vertex of the result with either a single vertex argument or
           with $opt{ first_root }, otherwise a random vertex is chosen.

           NOTE: note that all the vertices might not be reachable from the selected (explicit or
           random) start vertex.

           The start vertex is be available as the graph attribute "SPT_Bellman_Ford_root").

           The result weights of vertices can be retrieved from the result graph by

                   my $w = $sptg->get_vertex_attribute($v, 'weight');

           The predecessor vertex of a vertex in the result graph can be retrieved by

                   my $u = $sptg->get_vertex_attribute($v, 'p');

           ("A successor vertex" cannot be retrieved as simply because a single vertex can have
           several successors.  You can first find the "neighbors()" vertices and then remove the
           predecessor vertex.)

           If you want to find the shortes path between two vertices, see "SP_Bellman_Ford".

       SSSP_Bellman_Ford
           Alias for SPT_Bellman_Ford.

       SP_Bellman_Ford
               @path = $g->SP_Bellman_Ford($u, $v)

           Return the vertices in the shortest path in the graph $g between the two vertices $u,
           $v.  If no path can be found, an empty list is returned.

           Uses SPT_Bellman_Ford().

       SPT_Bellman_Ford_clear_cache
               $g->SPT_Bellman_Ford_clear_cache

           See "Clearing cached results".

   All-Pairs Shortest Paths (APSP)
       For either a directed or an undirected graph, return the APSP object describing all the
       possible paths between any two vertices of the graph.  If no weight is defined for an
       edge, 1 (one) is assumed.

       Note that weight of 0 (zero) does not mean do not use this edge, it means essentially the
       opposite: an edge that has zero cost, an edge that makes the vertices the same.

       APSP_Floyd_Warshall
       all_pairs_shortest_paths
               my $apsp = $g->APSP_Floyd_Warshall(...);

           Return the all-pairs shortest path object computed from the graph using Floyd-
           Warshall's algorithm.  The length of a path between two vertices is the sum of weight
           attribute of the edges along the shortest path between the two vertices.  If no weight
           attribute name is specified explicitly

               $g->APSP_Floyd_Warshall(attribute_name => 'height');

           the attribute "weight" is assumed.

           If an edge has no defined weight attribute, the value of one is assumed when getting
           the attribute.

           Once computed, you can query the APSP object with

           path_length
                       my $l = $apsp->path_length($u, $v);

                   Return the length of the shortest path between the two vertices.

           path_vertices
                       my @v = $apsp->path_vertices($u, $v);

                   Return the list of vertices along the shortest path.

           path_predecessor
                      my $u = $apsp->path_predecessor($v);

                   Returns the predecessor of vertex $v in the all-pairs shortest paths.

           average_path_length
                       my $apl = $g->average_path_length; # All vertex pairs.

                       my $apl = $g->average_path_length($u); # From $u.
                       my $apl = $g->average_path_length($u, undef); # From $u.

                       my $apl = $g->average_path_length($u, $v); # From $u to $v.

                       my $apl = $g->average_path_length(undef, $v); # To $v.

                   Return the average (shortest) path length over all the vertex pairs of the
                   graph, from a vertex, between two vertices, and to a vertex.

           longest_path
                       my @lp = $g->longest_path;
                       my $lp = $g->longest_path;

                   In scalar context return the longest shortest path length over all the vertex
                   pairs of the graph.  In list context return the vertices along a longest
                   shortest path.  Note that there might be more than one such path; this
                   interface returns a random one of them.

                   NOTE: this returns the longest shortest path, not the longest path.

           diameter
           graph_diameter
                       my $gd = $g->diameter;

                   The longest path over all the vertex pairs is known as the graph diameter.

                   For an unconnected graph, single-vertex, or empty graph, returns "undef".

           shortest_path
                       my @sp = $g->shortest_path;
                       my $sp = $g->shortest_path;

                   In scalar context return the shortest length over all the vertex pairs of the
                   graph.  In list context return the vertices along a shortest path.  Note that
                   there might be more than one such path; this interface returns a random one of
                   them.

                   For an unconnected, single-vertex, or empty graph, returns "undef" or an empty
                   list.

           radius
                       my $gr = $g->radius;

                   The shortest longest path over all the vertex pairs is known as the graph
                   radius.  See also "diameter".

                   For an unconnected, single-vertex, or empty graph, returns Infinity.

           center_vertices
           centre_vertices
                       my @c = $g->center_vertices;
                       my @c = $g->center_vertices($delta);

                   The graph center is the set of vertices for which the vertex eccentricity is
                   equal to the graph radius.  The vertices are returned in random order.  By
                   specifying a delta value you can widen the criterion from strict equality
                   (handy for non-integer edge weights).

                   For an unconnected, single-vertex, or empty graph, returns an empty list.

           vertex_eccentricity
                       my $ve = $g->vertex_eccentricity($v);

                   The longest path to a vertex is known as the vertex eccentricity.

                   If the graph is unconnected, single-vertex, or empty graph, returns Inf.

           You can walk through the matrix of the shortest paths by using

           for_shortest_paths
                   $n = $g->for_shortest_paths($callback)

               The number of shortest paths is returned (this should be equal to V*V).  The
               $callback is a sub reference that receives four arguments: the transitive closure
               object from Graph::TransitiveClosure, the two vertices, and the index to the
               current shortest paths (0..V*V-1).

   Clearing cached results
       For many graph algorithms there are several different but equally valid results.
       (Pseudo)Randomness is used internally by the Graph module to for example pick a random
       starting vertex, and to select random edges from a vertex.

       For efficiency the computed result is often cached to avoid recomputing the potentially
       expensive operation, and this also gives additional determinism (once a correct result has
       been computed, the same result will always be given).

       However, sometimes the exact opposite is desireable, and the possible alternative results
       are wanted (within the limits of the pseudorandomness: not all the possible solutions are
       guaranteed to be returned, usually only a subset is retuned).  To undo the caching, the
       following methods are available:

       ·   connectivity_clear_cache

           Affects "connected_components", "connected_component_by_vertex",
           "connected_component_by_index", "same_connected_components", "connected_graph",
           "is_connected", "is_weakly_connected", "weakly_connected_components",
           "weakly_connected_component_by_vertex", "weakly_connected_component_by_index",
           "same_weakly_connected_components", "weakly_connected_graph".

       ·   biconnectivity_clear_cache

           Affects "biconnected_components", "biconnected_component_by_vertex",
           "biconnected_component_by_index", "is_edge_connected", "is_edge_separable",
           "articulation_points", "cut_vertices", "is_biconnected", "biconnected_graph",
           "same_biconnected_components", "bridges".

       ·   strong_connectivity_clear_cache

           Affects "strongly_connected_components", "strongly_connected_component_by_vertex",
           "strongly_connected_component_by_index", "same_strongly_connected_components",
           "is_strongly_connected", "strongly_connected", "strongly_connected_graph".

       ·   SPT_Dijkstra_clear_cache

           Affects "SPT_Dijkstra", "SSSP_Dijkstra", "single_source_shortest_paths",
           "SP_Dijkstra".

       ·   SPT_Bellman_Ford_clear_cache

           Affects "SPT_Bellman_Ford", "SSSP_Bellman_Ford", "SP_Bellman_Ford".

       Note that any such computed and cached results are of course always automatically
       discarded whenever the graph is modified.

   Random
       You can either ask for random elements of existing graphs or create random graphs.

       random_vertex
               my $v = $g->random_vertex;

           Return a random vertex of the graph, or undef if there are no vertices.

       random_edge
               my $e = $g->random_edge;

           Return a random edge of the graph as an array reference having the vertices as
           elements, or undef if there are no edges.

       random_successor
               my $v = $g->random_successor($v);

           Return a random successor of the vertex in the graph, or undef if there are no
           successors.

       random_predecessor
               my $u = $g->random_predecessor($v);

           Return a random predecessor of the vertex in the graph, or undef if there are no
           predecessors.

       random_graph
               my $g = Graph->random_graph(%opt);

           Construct a random graph.  The %opt must contain the "vertices" argument

               vertices => vertices_def

           where the vertices_def is one of

           ·       an array reference where the elements of the array reference are the vertices

           ·       a number N in which case the vertices will be integers 0..N-1

       The %opt may have either of the argument "edges" or the argument "edges_fill".  Both are
       used to define how many random edges to add to the graph; "edges" is an absolute number,
       while "edges_fill" is a relative number (relative to the number of edges in a complete
       graph, C).  The number of edges can be larger than C, but only if the graph is countedged.
       The random edges will not include self-loops.  If neither "edges" nor "edges_fill" is
       specified, an "edges_fill" of 0.5 is assumed.

       If you want repeatable randomness (what is an oxymoron?)  you can use the "random_seed"
       option:

           $g = Graph->random_graph(vertices => 10, random_seed => 1234);

       As this uses the standard Perl srand(), the usual caveat applies: use it sparingly, and
       consider instead using a single srand() call at the top level of your application.

       The default random distribution of edges is flat, that is, any pair of vertices is equally
       likely to appear.  To define your own distribution, use the "random_edge" option:

           $g = Graph->random_graph(vertices => 10, random_edge => \&d);

       where "d" is a code reference receiving ($g, $u, $v, $p) as parameters, where the $g is
       the random graph, $u and $v are the vertices, and the $p is the probability ([0,1]) for a
       flat distribution.  It must return a probability ([0,1]) that the vertices $u and $v have
       an edge between them.  Note that returning one for a particular pair of vertices doesn't
       guarantee that the edge will be present in the resulting graph because the required number
       of edges might be reached before that particular pair is tested for the possibility of an
       edge.  Be very careful to adjust also "edges" or "edges_fill" so that there is a
       possibility of the filling process terminating.

       NOTE: a known problem with randomness in openbsd pre-perl-5.20 is that using a seed does
       not give you deterministic randomness. This affects any Perl code, not just Graph.

   Attributes
       You can attach free-form attributes (key-value pairs, in effect a full Perl hash) to each
       vertex, edge, and the graph itself.

       Note that attaching attributes does slow down some other operations on the graph by a
       factor of three to ten.  For example adding edge attributes does slow down anything that
       walks through all the edges.

       For vertex attributes:

       set_vertex_attribute
               $g->set_vertex_attribute($v, $name, $value)

           Set the named vertex attribute.

           If the vertex does not exist, the set_...() will create it, and the other vertex
           attribute methods will return false or empty.

           NOTE: any attributes beginning with an underscore/underline (_) are reserved for the
           internal use of the Graph module.

       get_vertex_attribute
               $value = $g->get_vertex_attribute($v, $name)

           Return the named vertex attribute.

       has_vertex_attribute
               $g->has_vertex_attribute($v, $name)

           Return true if the vertex has an attribute, false if not.

       delete_vertex_attribute
               $g->delete_vertex_attribute($v, $name)

           Delete the named vertex attribute.

       set_vertex_attributes
               $g->set_vertex_attributes($v, $attr)

           Set all the attributes of the vertex from the anonymous hash $attr.

           NOTE: any attributes beginning with an underscore ("_") are reserved for the internal
           use of the Graph module.

       get_vertex_attributes
               $attr = $g->get_vertex_attributes($v)

           Return all the attributes of the vertex as an anonymous hash.

       get_vertex_attribute_names
               @name = $g->get_vertex_attribute_names($v)

           Return the names of vertex attributes.

       get_vertex_attribute_values
               @value = $g->get_vertex_attribute_values($v)

           Return the values of vertex attributes.

       has_vertex_attributes
               $g->has_vertex_attributes($v)

           Return true if the vertex has any attributes, false if not.

       delete_vertex_attributes
               $g->delete_vertex_attributes($v)

           Delete all the attributes of the named vertex.

       If you are using multivertices, use the by_id variants:

       set_vertex_attribute_by_id
       get_vertex_attribute_by_id
       has_vertex_attribute_by_id
       delete_vertex_attribute_by_id
       set_vertex_attributes_by_id
       get_vertex_attributes_by_id
       get_vertex_attribute_names_by_id
       get_vertex_attribute_values_by_id
       has_vertex_attributes_by_id
       delete_vertex_attributes_by_id
               $g->set_vertex_attribute_by_id($v, $id, $name, $value)
               $g->get_vertex_attribute_by_id($v, $id, $name)
               $g->has_vertex_attribute_by_id($v, $id, $name)
               $g->delete_vertex_attribute_by_id($v, $id, $name)
               $g->set_vertex_attributes_by_id($v, $id, $attr)
               $g->get_vertex_attributes_by_id($v, $id)
               $g->get_vertex_attribute_values_by_id($v, $id)
               $g->get_vertex_attribute_names_by_id($v, $id)
               $g->has_vertex_attributes_by_id($v, $id)
               $g->delete_vertex_attributes_by_id($v, $id)

       For edge attributes:

       set_edge_attribute
               $g->set_edge_attribute($u, $v, $name, $value)

           Set the named edge attribute.

           If the edge does not exist, the set_...() will create it, and the other edge attribute
           methods will return false or empty.

           NOTE: any attributes beginning with an underscore ("_") are reserved for the internal
           use of the Graph module.

       get_edge_attribute
               $value = $g->get_edge_attribute($u, $v, $name)

           Return the named edge attribute.

       has_edge_attribute
               $g->has_edge_attribute($u, $v, $name)

           Return true if the edge has an attribute, false if not.

       delete_edge_attribute
               $g->delete_edge_attribute($u, $v, $name)

           Delete the named edge attribute.

       set_edge_attributes
               $g->set_edge_attributes($u, $v, $attr)

           Set all the attributes of the edge from the anonymous hash $attr.

           NOTE: any attributes beginning with an underscore ("_") are reserved for the internal
           use of the Graph module.

       get_edge_attributes
               $attr = $g->get_edge_attributes($u, $v)

           Return all the attributes of the edge as an anonymous hash.

       get_edge_attribute_names
               @name = $g->get_edge_attribute_names($u, $v)

           Return the names of edge attributes.

       get_edge_attribute_values
               @value = $g->get_edge_attribute_values($u, $v)

           Return the values of edge attributes.

       has_edge_attributes
               $g->has_edge_attributes($u, $v)

           Return true if the edge has any attributes, false if not.

       delete_edge_attributes
               $g->delete_edge_attributes($u, $v)

           Delete all the attributes of the named edge.

       If you are using multiedges, use the by_id variants:

       set_edge_attribute_by_id
       get_edge_attribute_by_id
       has_edge_attribute_by_id
       delete_edge_attribute_by_id
       set_edge_attributes_by_id
       get_edge_attributes_by_id
       get_edge_attribute_names_by_id
       get_edge_attribute_values_by_id
       has_edge_attributes_by_id
       delete_edge_attributes_by_id
               $g->set_edge_attribute_by_id($u, $v, $id, $name, $value)
               $g->get_edge_attribute_by_id($u, $v, $id, $name)
               $g->has_edge_attribute_by_id($u, $v, $id, $name)
               $g->delete_edge_attribute_by_id($u, $v, $id, $name)
               $g->set_edge_attributes_by_id($u, $v, $id, $attr)
               $g->get_edge_attributes_by_id($u, $v, $id)
               $g->get_edge_attribute_values_by_id($u, $v, $id)
               $g->get_edge_attribute_names_by_id($u, $v, $id)
               $g->has_edge_attributes_by_id($u, $v, $id)
               $g->delete_edge_attributes_by_id($u, $v, $id)

       For graph attributes:

       set_graph_attribute
               $g->set_graph_attribute($name, $value)

           Set the named graph attribute.

           NOTE: any attributes beginning with an underscore ("_") are reserved for the internal
           use of the Graph module.

       get_graph_attribute
               $value = $g->get_graph_attribute($name)

           Return the named graph attribute.

       has_graph_attribute
               $g->has_graph_attribute($name)

           Return true if the graph has an attribute, false if not.

       delete_graph_attribute
               $g->delete_graph_attribute($name)

           Delete the named graph attribute.

       set_graph_attributes
               $g->get_graph_attributes($attr)

           Set all the attributes of the graph from the anonymous hash $attr.

           NOTE: any attributes beginning with an underscore ("_") are reserved for the internal
           use of the Graph module.

       get_graph_attributes
               $attr = $g->get_graph_attributes()

           Return all the attributes of the graph as an anonymous hash.

       get_graph_attribute_names
               @name = $g->get_graph_attribute_names()

           Return the names of graph attributes.

       get_graph_attribute_values
               @value = $g->get_graph_attribute_values()

           Return the values of graph attributes.

       has_graph_attributes
               $g->has_graph_attributes()

           Return true if the graph has any attributes, false if not.

       delete_graph_attributes
               $g->delete_graph_attributes()

           Delete all the attributes of the named graph.

   Weighted
       As convenient shortcuts the following methods add, query, and manipulate the attribute
       "weight" with the specified value to the respective Graph elements.

       add_weighted_edge
               $g->add_weighted_edge($u, $v, $weight)

       add_weighted_edges
               $g->add_weighted_edges($u1, $v1, $weight1, ...)

       add_weighted_path
               $g->add_weighted_path($v1, $weight1, $v2, $weight2, $v3, ...)

       add_weighted_vertex
               $g->add_weighted_vertex($v, $weight)

       add_weighted_vertices
               $g->add_weighted_vertices($v1, $weight1, $v2, $weight2, ...)

       delete_edge_weight
               $g->delete_edge_weight($u, $v)

       delete_vertex_weight
               $g->delete_vertex_weight($v)

       get_edge_weight
               $g->get_edge_weight($u, $v)

       get_vertex_weight
               $g->get_vertex_weight($v)

       has_edge_weight
               $g->has_edge_weight($u, $v)

       has_vertex_weight
               $g->has_vertex_weight($v)

       set_edge_weight
               $g->set_edge_weight($u, $v, $weight)

       set_vertex_weight
               $g->set_vertex_weight($v, $weight)

   Isomorphism
       Two graphs being isomorphic means that they are structurally the same graph, the
       difference being that the vertices might have been renamed or substituted.  For example in
       the below example $g0 and $g1 are isomorphic: the vertices "b c d" have been renamed as "z
       x y".

               $g0 = Graph->new;
               $g0->add_edges(qw(a b a c c d));
               $g1 = Graph->new;
               $g1->add_edges(qw(a x x y a z));

       In the general case determining isomorphism is NP-hard, in other words, really hard (time-
       consuming), no other ways of solving the problem are known than brute force check of of
       all the possibilities (with possible optimization tricks, of course, but brute force still
       rules at the end of the day).

       A very rough guess at whether two graphs could be isomorphic is possible via the method

       could_be_isomorphic
               $g0->could_be_isomorphic($g1)

       If the graphs do not have the same number of vertices and edges, false is returned.  If
       the distribution of in-degrees and out-degrees at the vertices of the graphs does not
       match, false is returned.  Otherwise, true is returned.

       What is actually returned is the maximum number of possible isomorphic graphs between the
       two graphs, after the above sanity checks have been conducted.  It is basically the
       product of the factorials of the absolute values of in-degrees and out-degree pairs at
       each vertex, with the isolated vertices ignored (since they could be reshuffled and
       renamed arbitrarily).  Note that for large graphs the product of these factorials can
       overflow the maximum presentable number (the floating point number) in your computer (in
       Perl) and you might get for example Infinity as the result.

   Miscellaneous
       betweenness
               %b = $g->betweenness

           Returns a map of vertices to their Freeman's betweennesses:

             C_b(v) = \sum_{s \neq v \neq t \in V} \frac{\sigma_{s,t}(v)}{\sigma_{s,t}}

           It is described in:

               Freeman, A set of measures of centrality based on betweenness, http://arxiv.org/pdf/cond-mat/0309045

           and based on the algorithm from:

               "A Faster Algorithm for Betweenness Centrality"

       clustering_coefficient
               $gamma = $g->clustering_coefficient()
               ($gamma, %clustering) = $g->clustering_coefficient()

           Returns the clustering coefficient gamma as described in

               Duncan J. Watts and Steven Strogatz, Collective dynamics of 'small-world' networks, http://audiophile.tam.cornell.edu/SS_nature_smallworld.pdf

           In scalar context returns just the average gamma, in list context returns the average
           gamma and a hash of vertices to clustering coefficients.

       subgraph_by_radius
               $s = $g->subgraph_by_radius($n, $radius);

           Returns a subgraph representing the ball of $radius around node $n (breadth-first
           search).

       The "expect" methods can be used to test a graph and croak if the graph call is not as
       expected.

       expect_acyclic
       expect_dag
       expect_directed
       expect_hyperedged
       expect_hypervertexed
       expect_multiedged
       expect_multivertexed
       expect_no_args
       expect_non_multiedged
       expect_non_multivertexed
       expect_non_unionfind
       expect_undirected

       In many algorithms it is useful to have a value representing the infinity.  The Graph
       provides (and itself uses):

       Infinity
           (Not exported, use Graph::Infinity explicitly)

   Size Requirements
       A graph takes up at least 1172 bytes of memory.

       A vertex takes up at least 100 bytes of memory.

       An edge takes up at least 400 bytes of memory.

       (A Perl scalar value takes 16 bytes, or 12 bytes if it's a reference.)

       These size approximations are very approximate and optimistic (they are based on
       total_size() of Devel::Size).  In real life many factors affect these numbers, for example
       how Perl is configured.  The numbers are for a 32-bit platform and for Perl 5.8.8.

       Roughly, the above numbers mean that in a megabyte of memory you can fit for example a
       graph of about 1000 vertices and about 2500 edges.

   Hyperedges, hypervertices, hypergraphs
       BEWARE: this is a rather thinly tested feature, and the theory is even less so.  Do not
       expect this to stay as it is (or at all) in future releases.

       NOTE: most usual graph algorithms (and basic concepts) break horribly (or at least will
       look funny) with these hyperthingies.  Caveat emptor.

       Hyperedges are edges that connect a number of vertices different from the usual two.

       Hypervertices are vertices that consist of a number of vertices different from the usual
       one.

       Note that for hypervertices there is an asymmetry: when adding hypervertices, the single
       vertices are also implicitly added.

       Hypergraphs are graphs with hyperedges.

       To enable hyperness when constructing Graphs use the "hyperedged" and "hypervertexed"
       attributes:

          my $h = Graph->new(hyperedged => 1, hypervertexed => 1);

       To add hypervertexes, either explicitly use more than one vertex (or, indeed, no vertices)
       when using add_vertex()

          $h->add_vertex("a", "b")
          $h->add_vertex()

       or implicitly with array references when using add_edge()

          $h->add_edge(["a", "b"], "c")
          $h->add_edge()

       Testing for existence and deletion of hypervertices and hyperedges works similarly.

       To test for hyperness of a graph use the

       is_hypervertexed
       hypervertexed
               $g->is_hypervertexed
               $g->hypervertexed

       is_hyperedged
       hyperedged
               $g->is_hyperedged
               $g->hyperedged

       Since hypervertices consist of more than one vertex:

       vertices_at
               $g->vertices_at($v)

       Return the vertices at the vertex.  This may return just the vertex or also other
       vertices.

       To go with the concept of undirected in normal (non-hyper) graphs, there is a similar
       concept of omnidirected (this is my own coinage, "all-directions") for hypergraphs, and
       you can naturally test for it by

       is_omnidirected
       omnidirected
       is_omniedged
       omniedged
              $g->is_omniedged

              $g->omniedged

              $g->is_omnidirected

              $g->omnidirected

           Return true if the graph is omnidirected (edges have no direction), false if not.

       You may be wondering why on earth did I make up this new concept, why didn't the
       "undirected" work for me?  Well, because of this:

          $g = Graph->new(hypervertexed => 1, omnivertexed => 1);

       That's right, vertices can be omni, too - and that is indeed the default.  You can turn it
       off and then $g->add_vertex(qw(a b)) no more means adding also the (hyper)vertex qw(b a).
       In other words, the "directivity" is orthogonal to (or independent of) the number of
       vertices in the vertex/edge.

       is_omnivertexed
       omnivertexed

       Another oddity that fell out of the implementation is the uniqueness attribute, that comes
       naturally in "uniqedged" and "uniqvertexed" flavours.  It does what it sounds like, to
       unique or not the vertices participating in edges and vertices (is the hypervertex qw(a b
       a) the same as the hypervertex qw(a b), for example).  Without too much explanation:

       is_uniqedged
       uniqedged
       is_uniqvertexed
       uniqvertexed

   Backward compatibility with Graph 0.2
       The Graph 0.2 (and 0.2xxxx) had the following features

       ·   vertices() always sorted the vertex list, which most of the time is unnecessary and
           wastes CPU.

       ·   edges() returned a flat list where the begin and end vertices of the edges were
           intermingled: every even index had an edge begin vertex, and every odd index had an
           edge end vertex.  This had the unfortunate consequence of "scalar(@e = edges)" being
           twice the number of edges, and complicating any algorithm walking through the edges.

       ·   The vertex list returned by edges() was sorted, the primary key being the edge begin
           vertices, and the secondary key being the edge end vertices.

       ·   The attribute API was oddly position dependent and dependent on the number of
           arguments.  Use ..._graph_attribute(), ..._vertex_attribute(), ..._edge_attribute()
           instead.

       In future releases of Graph (any release after 0.50) the 0.2xxxx compatibility will be
       removed.  Upgrade your code now.

       If you want to continue using these (mis)features you can use the "compat02" flag when
       creating a graph:

           my $g = Graph->new(compat02 => 1);

       This will change the vertices() and edges() appropriately.  This, however, is not
       recommended, since it complicates all the code using vertices() and edges().  Instead it
       is recommended that the vertices02() and edges02() methods are used.  The corresponding
       new style (unsorted, and edges() returning a list of references) methods are called
       vertices05() and edges05().

       To test whether a graph has the compatibility turned on

       is_compat02
       compat02
               $g->is_compat02
               $g->compat02

       The following are not backward compatibility methods, strictly speaking, because they did
       not exist before.

       edges02
           Return the edges as a flat list of vertices, elements at even indices being the start
           vertices and elements at odd indices being the end vertices.

       edges05
           Return the edges as a list of array references, each element containing the vertices
           of each edge.  (This is not a backward compatibility interface as such since it did
           not exist before.)

       vertices02
           Return the vertices in sorted order.

       vertices05
           Return the vertices in random order.

       For the attributes the recommended way is to use the new API.

       Do not expect new methods to work for compat02 graphs.

       The following compatibility methods exist:

       has_attribute
       has_attributes
       get_attribute
       get_attributes
       set_attribute
       set_attributes
       delete_attribute
       delete_attributes
           Do not use the above, use the new attribute interfaces instead.

       vertices_unsorted
           Alias for vertices() (or rather, vertices05()) since the vertices() now always returns
           the vertices in an unsorted order.  You can also use the unsorted_vertices import, but
           only with a true value (false values will cause an error).

       density_limits
               my ($sparse, $dense, $complete) = $g->density_limits;

           Return the "density limits" used to classify graphs as "sparse" or "dense".  The first
           limit is C/4 and the second limit is 3C/4, where C is the number of edges in a
           complete graph (the last "limit").

       density
               my $density = $g->density;

           Return the density of the graph, the ratio of the number of edges to the number of
           edges in a complete graph.

       vertex
               my $v = $g->vertex($v);

           Return the vertex if the graph has the vertex, undef otherwise.

       out_edges
       in_edges
       edges($v)
           This is now called edges_at($v).

   DIAGNOSTICS
       ·   Graph::...Map...: arguments X expected Y ...

           If you see these (more user-friendly error messages should have been triggered above
           and before these) please report any such occurrences, but in general you should be
           happy to see these since it means that an attempt to call something with a wrong
           number of arguments was caught in time.

       ·   Graph::add_edge: graph is not hyperedged ...

           Maybe you used add_weighted_edge() with only the two vertex arguments.

       ·   Not an ARRAY reference at lib/Graph.pm ...

           One possibility is that you have code based on Graph 0.2xxxx that assumes Graphs being
           blessed hash references, possibly also assuming that certain hash keys are available
           to use for your own purposes.  In Graph 0.50 none of this is true.  Please do not
           expect any particular internal implementation of Graphs.  Use inheritance and
           graph/vertex/edge attributes instead.

           Another possibility is that you meant to have objects (blessed references) as graph
           vertices, but forgot to use "refvertexed" (see "refvertexed") when creating the graph.

ACKNOWLEDGEMENTS

       All bad terminology, bugs, and inefficiencies are naturally mine, all mine, and not the
       fault of the below.

       Thanks to Nathan Goodman and Andras Salamon for bravely betatesting my pre-0.50 code.  If
       they missed something, that was only because of my fiendish code.

       The following literature for algorithms and some test cases:

       ·   Algorithms in C, Third Edition, Part 5, Graph Algorithms, Robert Sedgewick, Addison
           Wesley

       ·   Introduction to Algorithms, First Edition, Cormen-Leiserson-Rivest, McGraw Hill

       ·   Graphs, Networks and Algorithms, Dieter Jungnickel, Springer

SEE ALSO

       Persistent/Serialized graphs?  You want to read/write Graphs?  See the Graph::Reader and
       Graph::Writer in CPAN.

REPOSITORY

       <https://github.com/neilbowers/Graph>

AUTHOR

       Jarkko Hietaniemi jhi@iki.fi

       Now being maintained by Neil Bowers <neilb@cpan.org>

COPYRIGHT AND LICENSE

       Copyright (c) 1998-2014 Jarkko Hietaniemi.  All rights reserved.

       This is free software; you can redistribute it and/or modify it under the same terms as
       the Perl 5 programming language system itself.