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 to v[k] in a digraph (V, E) is a non-empty sequence v, v, ..., 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 = 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

• `G = graph()`
• `V1 = V2 = vertex()`

• `G = graph()`
• `V1 = V2 = vertex()`
• `Label = label()`

• `G = graph()`
• `E = edge()`
• `V1 = V2 = vertex()`
• `Label = label()`
• ```add_edge_err_rsn() =     {bad_edge, Path :: [vertex()]} | {bad_vertex, V :: vertex()}```

`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(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 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, `{error, {bad_edge, Path}}` is returned. If `G` already has an edge with value `E` connecting a different pair of vertices, `{error, {bad_edge, [V1, V2]}}` 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`.

• `G = graph()`

• `G = graph()`
• `V = vertex()`

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

• `G = graph()`
• `V = vertex()`
• `Label = label()`

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

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

• `G = graph()`
• `E = edge()`

Deletes edge `E` from digraph `G`.

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

• `G = graph()`
• `Edges = [edge()]`

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

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

• `G = graph()`
• `V1 = V2 = vertex()`

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, v, ..., 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 = graph()`
• `V = vertex()`

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

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

• `G = graph()`
• `Vertices = [vertex()]`

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

### delete(G) -> true

• `G = graph()`

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

• `G = graph()`
• `E = edge()`
• `V1 = V2 = vertex()`
• `Label = label()`

Returns `{E, V1, V2, Label}`, 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

• `G = graph()`
• `Edges = [edge()]`

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

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

• `G = graph()`
• `V = vertex()`
• `Edges = [edge()]`

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

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

• `G = graph()`
• `V = vertex()`
• `Vertices = [vertex(), ...]`

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

• `G = graph()`
• `V1 = V2 = vertex()`
• `Vertices = [vertex(), ...]`

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

• `G = graph()`
• `V = vertex()`
• `Vertices = [vertex(), ...]`

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 `[V, V]`.

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

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

• `G = graph()`
• `V1 = V2 = vertex()`
• `Vertices = [vertex(), ...]`

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

• `G = graph()`
• `V = vertex()`

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

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

• `G = graph()`
• `V = vertex()`
• `Edges = [edge()]`

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

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

• `G = graph()`
• `V = vertex()`
• `Vertex = [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()

• `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`

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

• `G = graph()`

Returns the number of edges of digraph `G`.

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

• `G = graph()`

Returns the number of vertices of digraph `G`.

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

• `G = graph()`
• `V = vertex()`

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

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

• `G = graph()`
• `V = vertex()`
• `Edges = [edge()]`

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

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

• `G = graph()`
• `V = vertex()`
• `Vertices = [vertex()]`

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

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

• `G = graph()`
• `V = vertex()`
• `Label = label()`

Returns `{V, Label}`, 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

• `G = graph()`
• `Vertices = [vertex()]`

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