digraph
Directed Graphs
The digraph
module implements a version of labeled
directed graphs. What makes the graphs implemented here
nonproper 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 × 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 outdegree of a vertex is the number of edges emanating from that vertex. The indegree 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 outneighbour of v, and v is said to be an inneighbour of w. A path P from v[1] to v[k] in a digraph (V, E) is a nonempty 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 k1. 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.
Functions
add_edge(G, E, V1, V2, Label) > edge()  {error, Reason}
add_edge(G, V1, V2, Label) > edge()  {error, Reason}
add_edge(G, V1, V2) > edge()  {error, Reason}
G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()
Reason = {bad_edge, Path}  {bad_vertex, V}
Path = [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 ['$e'  N]
, where N
is an integer >= 0.
add_edge(G, V1, V2)
is equivalent to
add_edge(G, V1, V2, [])
.
If the edge would create a cycle in
an acyclic digraph,
then {error, {bad_edge, Path}}
is returned. If
either of V1
or V2
is not a vertex of the
digraph G
, then
{error, {bad_vertex,
V}}
is
returned, V = V1
or
V = V2
.
add_vertex(G, V, Label) > vertex()
add_vertex(G, V) > vertex()
add_vertex(G) > vertex()
G = digraph()
V = vertex()
Label = label()
add_vertex/3
creates (or modifies) the vertex V
of the digraph G
, using Label
as the (new)
label of the
vertex. Returns V
.
add_vertex(G, V)
is equivalent to
add_vertex(G, V, [])
.
add_vertex/1
creates a vertex using the empty list
as label, and returns the created vertex. The created vertex
is represented by the term ['$v'  N]
,
where N is an integer >= 0.
del_edge(G, E) > true
G = digraph()
E = edge()
Deletes the edge E
from the digraph G
.
del_edges(G, Edges) > true
G = digraph()
Edges = [edge()]
Deletes the edges in the list Edges
from the digraph
G
.
del_path(G, V1, V2) > true
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
G = digraph()
V = vertex()
del_vertices(G, Vertices) > true
G = digraph()
Vertices = [vertex()]
Deletes the vertices in the list Vertices
from the
digraph G
.
delete(G) > true
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
G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()
edges(G) > Edges
G = digraph()
Edges = [edge()]
Returns a list of all edges of the digraph G
, in
some unspecified order.
edges(G, V) > Edges
G = digraph()
V = vertex()
Edges = [edge()]
get_cycle(G, V) > Vertices  false
G = digraph()
V1 = V2 = 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
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 depthfirst manner,
and the first path found is returned.
get_short_cycle(G, V) > Vertices  false
G = digraph()
V1 = V2 = 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
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 breadthfirst
manner, and the first path found is returned.
in_degree(G, V) > integer()
G= digraph()
V = vertex()
Returns the indegree of the vertex
V
of the digraph G
.
in_edges(G, V) > Edges
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) > Vertices
G = digraph()
V = vertex()
Vertices = [vertex()]
Returns a list of
all inneighbours of
V
of the digraph G
, in some unspecified order.
info(G) > InfoList
G = digraph()
InfoList = [{cyclicity, Cyclicity}, {memory, NoWords}, {protection, Protection}]
Cyclicity = cyclic  acyclic
Protection = protected  private
NoWords = integer() >= 0
Returns a list of {Tag, Value}
pairs describing the
digraph G
. The following pairs are returned:

{cyclicity, Cyclicity}
, whereCyclicity
iscyclic
oracyclic
, according to the options given tonew
. 
{memory, NoWords}
, whereNoWords
is the number of words allocated to theets
tables. 
{protection, Protection}
, whereProtection
isprotected
orprivate
, according to the options given tonew
.
new() > digraph()
Equivalent to new([])
.
new(Type) > digraph()
Type = [cyclic  acyclic  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 be a badarg
exception.
no_edges(G) > integer() >= 0
G = digraph()
Returns the number of edges of the digraph G
.
no_vertices(G) > integer() >= 0
G = digraph()
Returns the number of vertices of the digraph G
.
out_degree(G, V) > integer()
G = digraph()
V = vertex()
Returns the outdegree of the vertex
V
of the digraph G
.
out_edges(G, V) > Edges
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
G = digraph()
V = vertex()
Vertices = [vertex()]
Returns a list of
all outneighbours of
V
of the digraph G
, in some unspecified order.
vertex(G, V) > {V, Label}  false
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
G = digraph()
Vertices = [vertex()]
Returns a list of all vertices of the digraph G
, in
some unspecified order.
See Also
 add_edge/5
 add_edge/4
 add_edge/3
 add_vertex/3
 add_vertex/2
 add_vertex/1
 del_edge/2
 del_edges/2
 del_path/3
 del_vertex/2
 del_vertices/2
 delete/1
 edge/2
 edges/1
 edges/2
 get_cycle/2
 get_path/3
 get_short_cycle/2
 get_short_path/3
 in_degree/2
 in_edges/2
 in_neighbours/2
 info/1
 new/0
 new/1
 no_edges/1
 no_vertices/1
 out_degree/2
 out_edges/2
 out_neighbours/2
 vertex/2
 vertices/1