digraph

## Directed graphs.

This module provides a version of labeled directed graphs. What makes the graphs provided here non-proper directed graphs is that multiple edges between vertices are allowed. However, the customary definition of directed graphs is used here.

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 more information. Such information can be attached to the vertices and to the edges of the digraph. An annotated digraph 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 an edge is emanating from v and incident on w, then w is said to be an out-neighbor of v, and v is said to be an in-neighbor 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 path P is k-1.

Path P is simple if all vertices are distinct, except that the first and the last vertices can be the same.

Path 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 without cycles.

### graph()

A digraph as returned by new/0,1.

### vertex()

#### Functions

add_edge/5 creates (or modifies) edge E of 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(GV1V2Label) is equivalent to add_edge(GEV1V2Label), where E is a created edge. The created edge is represented by term ['\$e' | N], where N is an integer >= 0.

If the edge would create a cycle in an acyclic digraph, {error, {bad_edge, Path}} is returned. If G already has an edge with value E connecting a different pair of vertices, {error, {bad_edge, [V1V2]}} is returned. If either of V1 or V2 is not a vertex of digraph G, {error, {bad_vertex, V}} is returned, V = V1 or V = V2.

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

add_vertex/3 creates (or modifies) vertex V of 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 term ['\$v' | N], where N is an integer >= 0.

### del_edge(G, E) -> true

Deletes edge E from digraph G.

### del_edges(G, Edges) -> true

Deletes the edges in list Edges from digraph G.

### del_path(G, V1, V2) -> true

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

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

### del_vertices(G, Vertices) -> true

Deletes the vertices in list Vertices from digraph G.

### delete(G) -> true

Deletes digraph G. This call is important as digraphs are implemented with ETS. There is no garbage collection of ETS tables. However, the digraph is deleted if the process that created the digraph terminates.

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

Returns {EV1V2Label}, where Label is the label of edge E emanating from V1 and incident on V2 of digraph G. If no edge E of digraph G exists, false is returned.

### edges(G) -> Edges

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

### edges(G, V) -> Edges

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

### get_cycle(G, V) -> Vertices | false

If a simple cycle of length two or more exists through vertex V, the cycle is returned as a list [V, ..., V] of vertices. If a loop through V exists, the loop is returned as a list [V]. If no cycles through V exist, false is returned.

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

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

Tries to find a simple path from vertex V1 to vertex V2 of 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.

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

### get_short_cycle(G, V) -> Vertices | false

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

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

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

Tries to find an as short as possible simple path from vertex V1 to vertex V2 of 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.

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

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

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

### in_edges(G, V) -> Edges

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

### in_neighbours(G, V) -> Vertex

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

### info(G) -> InfoList

• G = graph()
• 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 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()

• d_cyclicity() = acyclic | cyclic
• d_protection() = private | protected

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

cyclic

Allows 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 specified or Type is not a proper list, a badarg exception is raised.

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

Returns the number of edges of digraph G.

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

Returns the number of vertices of digraph G.

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

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

### out_edges(G, V) -> Edges

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

### out_neighbours(G, V) -> Vertices

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

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

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

### vertices(G) -> Vertices

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