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digraph(3erl) | Erlang Module Definition | digraph(3erl) |

# 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.

# DATA TYPES¶

d_type()=d_cyclicity()|d_protection()d_cyclicity()= acyclic | cyclicd_protection()= private | protectedgraph()

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, V1, V2, Label) -> edge() | {error, add_edge_err_rsn()}

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

Types:

G =
E =
V1 = V2 =
Label =

**graph()****edge()****vertex()****label()**add_edge_err_rsn()= {bad_edge, Path :: [vertex()]}| {bad_vertex, V ::vertex()}

*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.

*add_edge(G, V1, V2)*is equivalent to

*add_edge(G, V1, V2, [])*. 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*.

add_vertex(G) -> vertex()

add_vertex(G, V) -> vertex()

add_vertex(G, V, Label) -> vertex()

Types:

G =
V =
Label =

**graph()****vertex()****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(G, V)*is equivalent to

*add_vertex(G, 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 =
E =

**graph()****edge()**
Deletes the edge

*E*from the digraph*G*.del_edges(G, Edges) -> true

Types:

G =
Edges = [

**graph()****edge()**]
Deletes the edges in the list

*Edges*from the digraph*G*.del_path(G, V1, V2) -> true

Types:

G =
V1 = V2 =

**graph()****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 =
V =

**graph()****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 =
Vertices = [

**graph()****vertex()**]
Deletes the vertices in the list

*Vertices*from the digraph*G*.delete(G) -> true

Types:

G =

**graph()**
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 =
E =
V1 = V2 =
Label =

**graph()****edge()****vertex()****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 =
Edges = [

**graph()****edge()**]
Returns a list of all edges of the digraph

*G*, in some unspecified order.edges(G, V) -> Edges

Types:

G =
V =
Edges = [

**graph()****vertex()****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 =
V =
Vertices = [

**graph()****vertex()****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 =
V1 = V2 =
Vertices = [

**graph()****vertex()****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 =
V =
Vertices = [

**graph()****vertex()****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 =
V1 = V2 =
Vertices = [

**graph()****vertex()****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 =
V =

**graph()****vertex()**
Returns the

**in-degree**of the vertex*V*of the digraph*G*.in_edges(G, V) -> Edges

Types:

G =
V =
Edges = [

**graph()****vertex()****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 =
V =
Vertex = [

**graph()****vertex()****vertex()**]
Returns a list of all

**in-neighbours**of*V*of the digraph*G*, in some unspecified order.info(G) -> InfoList

Types:

G =
InfoList =

[{cyclicity, Cyclicity ::

{memory, NoWords :: integer() >= 0} |

{protection, Protection ::

**graph()**[{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() -> graph()

Equivalent to

*new([])*.new(Type) -> graph()

Types:

Type = [

**d_type()**]d_type()=d_cyclicity()|d_protection()

d_cyclicity()= acyclic | cyclic

d_protection()= private | protected

Returns an
If an unrecognized type option

**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.

*T*is given or*Type*is not a proper list, there will be a*badarg*exception.no_edges(G) -> integer() >= 0

Types:

G =

**graph()**
Returns the number of edges of the digraph

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

Types:

G =

**graph()**
Returns the number of vertices of the digraph

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

Types:

G =
V =

**graph()****vertex()**
Returns the

**out-degree**of the vertex*V*of the digraph*G*.out_edges(G, V) -> Edges

Types:

G =
V =
Edges = [

**graph()****vertex()****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 =
V =
Vertices = [

**graph()****vertex()****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 =
V =
Label =

**graph()****vertex()****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 =
Vertices = [

**graph()****vertex()**]
Returns a list of all vertices of the digraph

*G*, in some unspecified order.# SEE ALSO¶

**digraph_utils(3erl)**,

**ets(3erl)**

stdlib 2.2 | Ericsson AB |