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# 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 $g->is_undirected(). refvertexed refvertexed_stringified If you want to use references (including Perl objects) as vertices, use refvertexed. Note that using refvertexed means that internally the memory address of the reference (for example, a Perl object) is used as the "identifier" of the vertex, not the stringified form of the reference, even if you have defined your own stringification using overload. This avoids the problem of the stringified references potentially being identical (because they are identical in value, for example) even if the references are different. If you really want to use references and their stringified forms as the identities, use the refvertexed_stringified. But please do not stringify different objects to the same stringified value. unionfind If the graph is undirected, you can specify the unionfind parameter to use the so-called union-find scheme to speed up the computation of connected components of the graph (see "is_connected", "connected_components", "connected_component_by_vertex", "connected_component_by_index", and "same_connected_components"). If unionfind is used, adding edges (and vertices) becomes slower, but connectedness queries become faster. You must not delete edges or vertices of an unionfind graph, only add them. You can test a graph for "union-findness" with has_union_find Returns true if the graph was created with a true unionfind parameter. vertices An array reference of vertices to add. edges An array reference of array references of edge vertices to add. copy copy_graph  my$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;
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")

## 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.

   $g->add_path($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)

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

    $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. rename_vertex $g->rename_vertex('d', 'e')

Renames a vertex. It retains all of its edges. Throws exception if doesn't exist.

Returns the graph.

rename_vertices
    $g->rename_vertices(sub { uc$_[0] })

Calls a function for each vertex-name, renaming it to the return value.

Returns the graph.

ingest
    $g->ingest($g2)

Ingests all the vertices and edges of the given graph, including attributes. Returns the ingesting 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
    $g->is_connected For an undirected graph, return true is the graph is connected, false otherwise. Being connected means that from every vertex it is possible to reach every other vertex. If the graph has been created with a true unionfind parameter, the time complexity is (essentially) O(V), otherwise O(V log V). For directed graphs, see "is_strongly_connected" and "is_weakly_connected". connected_components  @cc =$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
    $g->connectivity_clear_cache See "Connected Graphs and Their Components" for further discussion. biconnectivity  my ($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
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".

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
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.

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 successors (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".

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

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

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.

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.

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 = $g->self_loop_vertices() Return the self-loop vertices of the graph. In scalar context return the number of self-loop vertices. See "is_self_loop_vertex" for the definition of a self-loop vertex. as_hashes  ($nodes, $edges) =$g->as_hashes

Return hash-refs which map vertices to their attributes, and for edges, a two-level hash mapping the predecessor to its successors, mapped to the attributes.

If multivertexed is true, the vertices hash will have the second-level values be the multivertex's ID, and the third level will be attributes as above.

If multiedged is true, similar will be true for the edges hash.

## 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

    $idv =$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:

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
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($v, 'p'); ("A successor vertex" cannot be retrieved as simply because a single vertex can have several successors. You can first find the neighbors() vertices and then remove the predecessor vertex.) If you want to find the shortest path between two vertices, see "SP_Bellman_Ford". SSSP_Bellman_Ford Alias for SPT_Bellman_Ford. SP_Bellman_Ford  @path =$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

## 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.

all_paths
    my @paths = $apsp->all_paths($u, $v); Return list of array-refs with all the paths from$u to $v. average_path_length  my$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 $v. Return the average (shortest) path length over all the non-zero paths between vertex pairs of the graph's transitive closure. Depending on the arguments, this can be from a vertex, between two vertices, or to a vertex. An undefined (or not-given) vertex will match all. longest_path  my @lp =$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 $callback is a sub reference that receives four arguments: the transitive closure object from Graph::TransitiveClosure, the two vertices, and the index to the current shortest paths (0..V*V-1). ## Clearing cached results For many graph algorithms there are several different but equally valid results. (Pseudo)Randomness is used internally by the Graph module to for example pick a random starting vertex, and to select random edges from a vertex. For efficiency the computed result is often cached to avoid recomputing the potentially expensive operation, and this also gives additional determinism (once a correct result has been computed, the same result will always be given). However, sometimes the exact opposite is desireable, and the possible alternative results are wanted (within the limits of the pseudorandomness: not all the possible solutions are guaranteed to be returned, usually only a subset is retuned). To undo the caching, the following methods are available: • connectivity_clear_cache • biconnectivity_clear_cache • strong_connectivity_clear_cache • SPT_Dijkstra_clear_cache • SPT_Bellman_Ford_clear_cache Note that any such computed and cached results are of course always automatically discarded whenever the graph is modified. ## Random You can either ask for random elements of existing graphs or create random graphs. random_vertex  my$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($v) Return all the attributes of the vertex as an anonymous hash, or undef if no such vertex. get_vertex_attribute_names  @name =$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, or undef if no such edge.

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)
    $g->add_weighted_vertices($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, https://web.archive.org/web/20120616204225/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.

Returns an empty list (and therefore undefined in scalar context) if the graph has no vertices.

    $s =$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

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

# AUTHOR

Jarkko Hietaniemi jhi@iki.fi

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