Graph - graph data structures and algorithms 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.

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
Create an empty graph. 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 If you want to use references (including Perl objects) as vertices. 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 can test a graph for "union-findness" with has_union_find vertices An array reference of vertices to add. edges An array reference of array references of edge vertices to add. 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 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). 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). 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). 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 $g->transpose_edge($u, $v) 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. 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). See also "random_graph" for a random constructor.

Basics
$g->add_vertex($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. $g->add_edge($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. $g->has_vertex($v) Return true if the vertex exists in the graph, false otherwise. $g->has_edge($u, $v) Return true if the edge exists in the graph, false otherwise. $g->delete_vertex($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. $g->delete_vertices($v1, $v2, ...) Delete the vertices from the graph. Returns 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. $g->delete_edge($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. $g->delete_edges($u1, $v1, $u2, $v2, ...) Delete the edges from the graph. Returns 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

Comparing
Testing for equality can be done either by the overloaded "eq" operator $g eq "a-b,a-c,d" or by the method $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 $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. $g->add_path($a, $b, $c, ..., $x, $y, $z) Add the edges $a-$b, $b-$c, ..., $x-$y, $y-$z to the graph. Returns the graph. $g->has_path($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. $g->delete_path($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. $g->add_cycle($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. $g->has_cycle($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". $g->delete_cycle($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. $g->has_a_cycle Returns true if the graph has a cycle, false if not. $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
$g->is_simple_graph Return true if the graph has no multiedges, false otherwise. $g->is_pseudo_graph Return true if the graph has any multiedges or any self-loops, false otherwise. $g->is_multi_graph Return true if the graph has any multiedges but no self-loops, false otherwise. $g->is_directed_acyclic_graph $g->is_dag Return true if the graph is directed and acyclic, false otherwise. $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 that one cycle, see "find_a_cycle". Return true if the graph is acyclic (does not contain any cycles). To find a cycle, use find_a_cycle.

Transitivity
$g->is_transitive Return true if the graph is transitive, false otherwise. $tcg = $g->TransitiveClosure_Floyd_Warshall Return the transitive closure graph of the graph. You can query the reachability from $u to $v with $tcg->is_reachable($u, $v) See Graph::TransitiveClosure for more information about creating and querying transitive closures. With $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. $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
$g->is_directed() $g->directed() Return true if the graph is directed, false otherwise. $g->is_undirected() $g->undirected() Return true if the graph is undirected, false otherwise. Return true if the graph can handle references (including Perl objects) as 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. $g->has_vertices() Return true if the graph has any vertices, false otherwise. 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. $g->has_edges() Return true if the graph has any edges, false otherwise. $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". @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". $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". @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". $g->same_connected_components($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". $cg = $g->connected_graph For an undirected graph, return its connected graph. $g->connectivity_clear_cache See "Clearing cached results". See "Connected Graphs and Their Components" for further discussion. my ($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. $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). $g->is_edge_connected For an undirected graph, return true if the graph is edge-connected (if it has no bridges). $g->is_edge_separable For an undirected graph, return true if the graph is edge-separable (if it has bridges). $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. $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. $i = $g->biconnected_component_by_index($v) For an undirected graph, return an index identifying the biconnected component the vertex belongs to, the indexing starting from zero. For the inverse, see "connected_component_by_index". For directed graphs, see "strongly_connected_component_by_index" and "weakly_connected_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. $g->same_biconnected_components($u, $v, ...) For an undirected graph, return true if the vertices are in the same biconnected component. $bcg = $g->biconnected_graph For an undirected graph, return its biconnected graph. See "Connected Graphs and Their Components" for further discussion. $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. $g->biconnectivity_clear_cache See "Clearing cached results". $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". $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". @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". $g->same_strongly_connected_components($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". $g->strong_connectivity_clear_cache See "Clearing cached results". $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". @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". $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". @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. $g->same_weakly_connected_components($u, $v, ...) Return true if the vertices are in the same weakly connected component. $wcg = $g->weakly_connected_graph For a directed graph, return its weakly connected graph. For undirected graphs, see "connected_graph" and "biconnected_graph". 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. See also "weakly_connected_components". For undirected graphs, see "connected_components", or see "biconnected_components". 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". $g->is_sink_vertex($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(). $g->is_source_vertex($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(). $g->is_successorless_vertex($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(). $g->is_successorful_vertex($v) Return true if the vertex $v has successors, false if not. $g->is_predecessorless_vertex($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(). $g->is_predecessorful_vertex($v) Return true if the vertex $v has predecessors, false if not. $g->is_isolated_vertex($v) Return true if the vertex $v is an isolated vertex: no successors and no predecessors. $g->is_interior_vertex($v) Return true if the vertex $v is an interior vertex: both successors and predecessors. $g->is_exterior_vertex($v) Return true if the vertex $v is an exterior vertex: has either no successors or no predecessors, or neither. $g->is_self_loop_vertex($v) Return true if the vertex $v is a self loop vertex: has an edge from itself to itself. @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. @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. @v = $g->successorful_vertices() Return the successorful vertices of the graph. In scalar context return the number of successorful vertices. @v = $g->successorless_vertices() Return the successorless vertices of the graph. In scalar context return the number of successorless vertices. @s = $g->successors($v) Return the immediate successor vertices of the vertex. See also "all_successors", "all_neighbours", and "all_reachable". @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". @n = $g->neighbours($v) Return the neighbo(u)ring vertices. Also known as the adjacent vertices. See also "all_neighbours" and "all_reachable". @n = $g->all_neighbours(@v) Return the neighbo(u)ring 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. See also "neighbours" and "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". See also "neighbours", "all_neighbours", and "all_successors". @v = $g->predecessorful_vertices() Return the predecessorful vertices of the graph. In scalar context return the number of predecessorful vertices. @v = $g->predecessorless_vertices() Return the predecessorless vertices of the graph. In scalar context return the number of predecessorless vertices. @p = $g->predecessors($v) Return the immediate predecessor vertices of the vertex. See also "all_predecessors", "all_neighbours", and "all_reachable". @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". @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. @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. @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. @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. $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()"). $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()"). $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()"). my $ad = $g->average_degree; Return the average degree (as in "degree()" or "vertex_degree()") taken over all vertices. Related methods are @e = $g->edges_at($v) The union of edges from and edges to at the vertex. @e = $g->edges_from($v) The edges leaving the vertex. @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: 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 $g->is_countedged $g->countedged Return true if the graph is countedged. $g->is_countvertexed $g->countvertexed Return true if the graph is countvertexed. $g->is_multiedged $g->multiedged Return true if the graph is multiedged. $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 @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: $g->add_vertex_by_id($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) 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 $g->get_multivertex_ids($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): 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:

Topological Sort
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. $mstg = $g->MST_Kruskal; Returns the Kruskal MST of the graph. $mstg = $g->MST_Prim(%opt); Returns the Prim MST of the graph. You can choose the first vertex with $opt{ first_root }. $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. $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. The 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". Aliases for SPT_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(). $g->SPT_Dijkstra_clear_cache See "Clearing cached results". $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". Alias for SPT_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(). $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. 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 my $l = $apsp->path_length($u, $v); Return the length of the shortest path between the two vertices. my @v = $apsp->path_vertices($u, $v); Return the list of vertices along the shortest path. my $u = $apsp->path_predecessor($v); Returns the predecessor of vertex $v in the all-pairs shortest paths. 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. 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 interfaces return a random one of them. my $gd = $g->diameter; The longest path over all the vertex pairs is known as the graph diameter. 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. my $gr = $g->radius; The shortest longest path over all the vertex pairs is known as the graph radius. See also "diameter". 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). my $ve = $g->vertex_eccentricity($v); The longest path to a vertex is known as the vertex eccentricity. If the graph is unconnected, returns Inf. You can walk through the matrix of the shortest paths by using $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. my $v = $g->random_vertex; Return a random vertex of the graph, or undef if there are no vertices. 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. my $v = $g->random_successor($v); Return a random successor of the vertex in the graph, or undef if there are no successors. my $u = $g->random_predecessor($v); Return a random predecessor of the vertex in the graph, or undef if there are no predecessors. 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.

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: $g->set_vertex_attribute($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. $value = $g->get_vertex_attribute($v, $name) Return the named vertex attribute. $g->has_vertex_attribute($v, $name) Return true if the vertex has an attribute, false if not. $g->delete_vertex_attribute($v, $name) Delete the named vertex attribute. $g->set_vertex_attributes($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. $attr = $g->get_vertex_attributes($v) Return all the attributes of the vertex as an anonymous hash. @name = $g->get_vertex_attribute_names($v) Return the names of vertex attributes. @value = $g->get_vertex_attribute_values($v) Return the values of vertex attributes. $g->has_vertex_attributes($v) Return true if the vertex has any attributes, false if not. $g->delete_vertex_attributes($v) Delete all the attributes of the named vertex. If you are using multivertices, use the by_id variants: $g->set_vertex_attribute_by_id($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: $g->set_edge_attribute($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. $value = $g->get_edge_attribute($u, $v, $name) Return the named edge attribute. $g->has_edge_attribute($u, $v, $name) Return true if the edge has an attribute, false if not. $g->delete_edge_attribute($u, $v, $name) Delete the named edge attribute. $g->set_edge_attributes($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. $attr = $g->get_edge_attributes($u, $v) Return all the attributes of the edge as an anonymous hash. @name = $g->get_edge_attribute_names($u, $v) Return the names of edge attributes. @value = $g->get_edge_attribute_values($u, $v) Return the values of edge attributes. $g->has_edge_attributes($u, $v) Return true if the edge has any attributes, false if not. $g->delete_edge_attributes($u, $v) Delete all the attributes of the named edge. If you are using multiedges, use the by_id variants: $g->set_edge_attribute_by_id($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: $g->set_graph_attribute($name, $value) Set the named graph attribute. NOTE: any attributes beginning with an underscore ("_") are reserved for the internal use of the Graph module. $value = $g->get_graph_attribute($name) Return the named graph attribute. $g->has_graph_attribute($name) Return true if the graph has an attribute, false if not. $g->delete_graph_attribute($name) Delete the named graph attribute. $g->get_graph_attributes($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. $attr = $g->get_graph_attributes() Return all the attributes of the graph as an anonymous hash. @name = $g->get_graph_attribute_names() Return the names of graph attributes. @value = $g->get_graph_attribute_values() Return the values of graph attributes. $g->has_graph_attributes() Return true if the graph has any attributes, false if not. $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. $g->add_weighted_edge($u, $v, $weight) $g->add_weighted_edges($u1, $v1, $weight1, ...) $g->add_weighted_path($v1, $weight1, $v2, $weight2, $v3, ...) $g->add_weighted_vertex($v, $weight) $g->add_weighted_vertices($v1, $weight1, $v2, $weight2, ...) $g->delete_edge_weight($u, $v) $g->delete_vertex_weight($v) $g->get_edge_weight($u, $v) $g->get_vertex_weight($v) $g->has_edge_weight($u, $v) $g->has_vertex_weight($v) $g->set_edge_weight($u, $v, $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 $g0->could_be_isomorphic($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
%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://links.jstor.org/sici?sici=0038-0431(197703)40%3A1%3C35%3AASOMOC%3E2.0.CO%3B2-H and based on the algorithm from: "A Faster Algorithm for Betweenness Centrality" $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://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. $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 is not as expected. In many algorithms it is useful to have a value representing the infinity. The Graph provides (and itself uses): (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 $g->is_hypervertexed $g->hypervertexed $g->is_hyperedged $g->hyperedged Since hypervertices consist of more than one vertex: $g->vertices_at($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 $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. 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:

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 $g->is_compat02 $g->compat02 The following are not backward compatibility methods, strictly speaking, because they did not exist before. Return the edges as a flat list of vertices, elements at even indices being the start vertices and elements at odd indices being the end vertices. Return the edges as a list of array references, each element containing the vertices of each edge. (This is not a backward compatibility interface as such since it did not exist before.) Return the vertices in sorted order. Return the vertices in random order. For the attributes the recommended way is to use the new API. Do not expect new methods to work for compat02 graphs. The following compatibility methods exist: Do not use the above, use the new attribute interfaces instead. Alias for vertices() (or rather, vertices05()) since the vertices() now always returns the vertices in an unsorted order. You can also use the unsorted_vertices import, but only with a true value (false values will cause an error). my ($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"). 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. my $v = $g->vertex($v); Return the vertex if the graph has the vertex, undef otherwise. 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.

POSSIBLE FUTURES
A possible future direction is a new graph module written for speed: this may very possibly mean breaking or limiting some of the APIs or behaviour as compared with this release of the module. What definitely won't happen in future releases is carrying over the Graph 0.2xxxx backward compatibility API. 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. Jarkko Hietaniemi jhi@iki.fi This module is licensed under the same terms as Perl itself.