# digraph

## Directed Graphs

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 × 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 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 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 = 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.

### 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()

• `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, 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 = 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

• `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()`

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

• `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()]`

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

• `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 depth-first 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 breadth-first manner, and the first path found is returned.

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

• `G= digraph()`
• `V = vertex()`

Returns the in-degree 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 in-neighbours 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}`, 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() | {error, Reason}

• `Type = [cyclic | acyclic | private | protected]`
• `Reason = {unknown_type, term()}`

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, then `{error, {unknown_type, `T`}}` is returned.

### 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 out-degree 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 out-neighbours 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.