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#### NAME

```       digraph - Directed Graphs

```

#### DESCRIPTION

```       The  digraph module implements a version of labeled directed graphs. What makes the graphs
implemented here non-proper directed graphs is that multiple edges  between  vertices  are
allowed.  However,  the  customary  definition of directed graphs will be used in the text
that follows.

A directed graph (or just "digraph") is a pair (V, E) of a finite set V of vertices and  a
finite  set E of directed edges (or just "edges"). The set of edges E is a subset of V x V
(the Cartesian product of V with itself). In this module, V is allowed to be empty; the so
obtained  unique  digraph  is  called  the  empty  digraph.  Both  vertices  and edges are
represented by unique Erlang terms.

Digraphs can be annotated with additional information. Such information may be attached to
the vertices and to the edges of the digraph. A digraph which has been annotated is called
a labeled digraph, and the information attached to a vertex or an edge is called a  label.
Labels are Erlang terms.

An  edge  e  = (v, w) is said to emanate from vertex v and to be incident on vertex w. The
out-degree of a vertex is the number of edges emanating from that vertex. The in-degree of
a  vertex  is  the  number of edges incident on that vertex. If there is an edge emanating
from v and incident on w, then w is said to be an out-neighbour of v, and v is said to  be
an  in-neighbour  of  w.  A  path  P  from v[1] to v[k] in a digraph (V, E) is a non-empty
sequence v[1], v[2], ..., v[k] of vertices in V such that there is an  edge  (v[i],v[i+1])
in  E  for  1  <=  i < k. The length of the path P is k-1. P is simple if all vertices are
distinct, except that the first and the last vertices may be the same. P is a cycle if the
length  of  P is not zero and v[1] = v[k]. A loop is a cycle of length one. A simple cycle
is a path that is both a cycle and simple. An acyclic digraph is a  digraph  that  has  no
cycles.

```

#### DATATYPES

```       d_type() = d_cyclicity() | d_protection()

d_cyclicity() = acyclic | cyclic

d_protection() = private | protected

digraph()

A digraph as returned by new/0,1.

edge()

label() = term()

vertex()

```

#### EXPORTS

```       add_edge(G, V1, V2) -> edge() | {error, add_edge_err_rsn()}

add_edge(G, E, V1, V2, Label) ->

Types:

G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()

add_edge/5  creates  (or  modifies) the edge E of the digraph G, using Label as the
(new) label of the edge. The edge is emanating from V1 and incident on V2.  Returns
E.

add_edge(G,  V1, V2, Label) is equivalent to add_edge(G, E, V1, V2, Label), where E
is a created edge. The created edge is represented by the term ['\$e' | N], where  N
is an integer >= 0.

If  the  edge  would  create a cycle in an acyclic digraph, then {error, {bad_edge,
Path}} is returned. If either of V1 or V2 is not a vertex of the  digraph  G,  then
{error, {bad_vertex, V}} is returned, V = V1 or V = V2.

Types:

G = digraph()
V = vertex()
Label = label()

add_vertex/3  creates  (or  modifies) the vertex V of the digraph G, using Label as
the (new) label of the vertex. Returns V.

add_vertex/1 creates a vertex using the  empty  list  as  label,  and  returns  the
created  vertex.  The created vertex is represented by the term ['\$v' | N], where N
is an integer >= 0.

del_edge(G, E) -> true

Types:

G = digraph()
E = edge()

Deletes the edge E from the digraph G.

del_edges(G, Edges) -> true

Types:

G = digraph()
Edges = [edge()]

Deletes the edges in the list Edges from the digraph G.

del_path(G, V1, V2) -> true

Types:

G = digraph()
V1 = V2 = vertex()

Deletes edges from the digraph G until there are no paths from the vertex V1 to the
vertex V2.

A  sketch of the procedure employed: Find an arbitrary simple path v[1], v[2], ...,
v[k] from V1 to V2 in G. Remove all edges of G emanating from v[i] and incident  to
v[i+1]  for  1  <=  i < k (including multiple edges). Repeat until there is no path
between V1 and V2.

del_vertex(G, V) -> true

Types:

G = digraph()
V = vertex()

Deletes the vertex V from the digraph G. Any edges emanating from V or incident  on
V are also deleted.

del_vertices(G, Vertices) -> true

Types:

G = digraph()
Vertices = [vertex()]

Deletes the vertices in the list Vertices from the digraph G.

delete(G) -> true

Types:

G = digraph()

Deletes the digraph G. This call is important because digraphs are implemented with
ETS. There is no garbage collection of ETS tables. The digraph  will,  however,  be
deleted if the process that created the digraph terminates.

edge(G, E) -> {E, V1, V2, Label} | false

Types:

G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()

Returns {E, V1, V2, Label} where Label is the label of the edge E emanating from V1
and incident on V2 of the digraph G. If there is no edge E of the digraph  G,  then
false is returned.

edges(G) -> Edges

Types:

G = digraph()
Edges = [edge()]

Returns a list of all edges of the digraph G, in some unspecified order.

edges(G, V) -> Edges

Types:

G = digraph()
V = vertex()
Edges = [edge()]

Returns  a  list  of all edges emanating from or incident on V of the digraph G, in
some unspecified order.

get_cycle(G, V) -> Vertices | false

Types:

G = digraph()
V = vertex()
Vertices = [vertex(), ...]

If there is a simple cycle of length two or more through the  vertex  V,  then  the
cycle  is  returned as a list [V, ..., V] of vertices, otherwise if there is a loop
through V, then the loop is returned as a list [V]. If there are no cycles  through
V, then false is returned.

get_path/3 is used for finding a simple cycle through V.

get_path(G, V1, V2) -> Vertices | false

Types:

G = digraph()
V1 = V2 = vertex()
Vertices = [vertex(), ...]

Tries  to  find a simple path from the vertex V1 to the vertex V2 of the digraph G.
Returns the path as a list [V1, ..., V2] of vertices, or false if  no  simple  path
from V1 to V2 of length one or more exists.

The  digraph  G  is  traversed in a depth-first manner, and the first path found is
returned.

get_short_cycle(G, V) -> Vertices | false

Types:

G = digraph()
V = vertex()
Vertices = [vertex(), ...]

Tries to find an as short as possible simple cycle through  the  vertex  V  of  the
digraph  G.  Returns  the  cycle  as a list [V, ..., V] of vertices, or false if no
simple cycle through V exists. Note that a loop through V is returned as  the  list
[V, V].

get_short_path/3 is used for finding a simple cycle through V.

get_short_path(G, V1, V2) -> Vertices | false

Types:

G = digraph()
V1 = V2 = vertex()
Vertices = [vertex(), ...]

Tries  to find an as short as possible simple path from the vertex V1 to the vertex
V2 of the digraph G. Returns the path as a list [V1, ..., V2] of vertices, or false
if no simple path from V1 to V2 of length one or more exists.

The  digraph  G is traversed in a breadth-first manner, and the first path found is
returned.

in_degree(G, V) -> integer() >= 0

Types:

G = digraph()
V = vertex()

Returns the in-degree of the vertex V of the digraph G.

in_edges(G, V) -> Edges

Types:

G = digraph()
V = vertex()
Edges = [edge()]

Returns a list of all edges incident on V of the digraph  G,  in  some  unspecified
order.

in_neighbours(G, V) -> Vertex

Types:

G = digraph()
V = vertex()
Vertex = [vertex()]

Returns  a  list  of  all  in-neighbours of V of the digraph G, in some unspecified
order.

info(G) -> InfoList

Types:

G = digraph()
InfoList =
[{cyclicity, Cyclicity :: d_cyclicity()} |
{memory, NoWords :: integer() >= 0} |
{protection, Protection :: d_protection()}]
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected

Returns a list of {Tag, Value} pairs describing the digraph G. The following  pairs
are returned:

* {cyclicity,  Cyclicity}, where Cyclicity is cyclic or acyclic, according to the
options given to new.

* {memory, NoWords}, where NoWords is the number of words allocated  to  the  ETS
tables.

* {protection,  Protection},  where Protection is protected or private, according
to the options given to new.

new() -> digraph()

Equivalent to new([]).

new(Type) -> digraph()

Types:

Type = [d_type()]
d_type() = d_cyclicity() | d_protection()
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected

Returns an empty digraph with properties according to the options in Type:

cyclic:
Allow cycles in the digraph (default).

acyclic:
The digraph is to be kept acyclic.

protected:
Other processes can read the digraph (default).

private:
The digraph can be read and modified by the creating process only.

If an unrecognized type option T is given or Type is not a proper list, there  will

no_edges(G) -> integer() >= 0

Types:

G = digraph()

Returns the number of edges of the digraph G.

no_vertices(G) -> integer() >= 0

Types:

G = digraph()

Returns the number of vertices of the digraph G.

out_degree(G, V) -> integer() >= 0

Types:

G = digraph()
V = vertex()

Returns the out-degree of the vertex V of the digraph G.

out_edges(G, V) -> Edges

Types:

G = digraph()
V = vertex()
Edges = [edge()]

Returns  a list of all edges emanating from V of the digraph G, in some unspecified
order.

out_neighbours(G, V) -> Vertices

Types:

G = digraph()
V = vertex()
Vertices = [vertex()]

Returns a list of all out-neighbours of V of the digraph  G,  in  some  unspecified
order.

vertex(G, V) -> {V, Label} | false

Types:

G = digraph()
V = vertex()
Label = label()

Returns  {V,  Label}  where Label is the label of the vertex V of the digraph G, or
false if there is no vertex V of the digraph G.

vertices(G) -> Vertices

Types:

G = digraph()
Vertices = [vertex()]

Returns a list of all vertices of the digraph G, in some unspecified order.

```

#### SEEALSO

```       digraph_utils(3erl), ets(3erl)
```