 # digraph

digraph

### MODULE SUMMARY

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 "graph") 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 graph is called the empty graph. Both vertices and edges are represented by unique Erlang terms.

Graphs can be annotated with additional information. Such information may be attached to the vertices and to the edges of the graph. A graph which has been annotated is called a labeled graph, 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 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 graph (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 graph is a graph that has no cycles.

### EXPORTS

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

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

`cyclic`
Allow cycles in the graph (default).
`acyclic`
The graph is to be kept acyclic.
`public`
The graph may be read and modified by any process.
`protected`
Other processes can only read the graph (default).
`private`
The graph 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.

Equivalent to `new([])`.

Types:
`G = graph()`

Deletes the graph `G`. This call is important because graphs are implemented with `ets`. There is no garbage collection of `ets` tables. The graph will, however, be deleted if the process that created the graph terminates.

Types:
`G = graph()`
```InfoList = [{cyclicity, Cyclicity}, {memory, NoWords}, {protection, Protection}]```
`Cyclicity = cyclic | acyclic`
`Protection = public | protected | private`
`NoWords = integer() >= 0`

Returns a list of `{Tag, Value}` pairs describing the graph `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 `public`, `protected` or `private`, according to the options given to `new`.

Types:
`G = graph()`
`V = vertex()`
`Label = label()`

`add_vertex/3` creates (or modifies) the vertex `V` of the graph `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. Tuples on the form `[´\$v´ | N]`, where N is an integer >= 1, are used for representing the created vertices.

Types:
`G = graph()`
`V = vertex()`
`Label = label()`

Returns `{V, Label}` where `Label` is the label of the vertex `V` of the graph `G`, or `false` if there is no vertex `V` of the graph `G`.

Types:
`G = graph()`

Returns the number of vertices of the graph `G`.

Types:
`G = graph()`
`Vertices = [vertex()]`

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

Types:
`G = graph()`
`V = vertex()`

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

Types:
`G = graph()`
`Vertices = [vertex()]`

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

Types:
`G = graph()`
`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 graph `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. Tuples on the form `[´\$e´ | N]`, where N is an integer >= 1, are used for representing the created edges.

`add_edge(G, V1, V2)` is equivalent to `add_edge(G, V1, V2, [])`.

If the edge would create a cycle in an acyclic graph, then `{error, {bad_edge, Path}}` is returned. If either of `V1` or `V2` is not a vertex of the graph `G`, then `{error, {bad_vertex, `V`}}` is returned, V = `V1` or V = `V2`.

Types:
`G = graph()`
`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 graph `G`. If there is no edge `E` of the graph `G`, then `false` is returned.

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

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

Types:
`G = graph()`

Returns the number of edges of the graph `G`.

Types:
`G = graph()`
`Edges = [edge()]`

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

Types:
`G = graph()`
`E = edge()`

Deletes the edge `E` from the graph `G`.

Types:
`G = graph()`
`Edges = [edge()]`

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

Types:
`G = graph()`
`V = vertex()`
`Vertices = [vertex()]`

Returns a list of all out-neighbours of `V` of the graph `G`, in some unspecified order.

Types:
`G = graph()`
`V = vertex()`
`Vertices = [vertex()]`

Returns a list of all in-neighbours of `V` of the graph `G`, in some unspecified order.

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

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

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

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

Types:
`G = graph()`
`V = vertex()`

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

Types:
`G= graph()`
`V = vertex()`

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

Types:
`G = graph()`
`V1 = V2 = vertex()`

Deletes edges from the graph `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`.

Types:
`G = graph()`
`V1 = V2 = vertex()`
`Vertices = [vertex()]`

Tries to find a simple path from the vertex `V1` to the vertex `V2` of the graph `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 graph `G` is traversed in a depth-first manner, and the first path found is returned.

Types:
`G = graph()`
`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 graph `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 graph `G` is traversed in a breadth-first manner, and the first path found is returned.

Types:
`G = graph()`
`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`.

Types:
`G = graph()`
`V1 = V2 = vertex()`
`Vertices = [vertex()]`

Tries to find an as short as possible simple cycle through the vertex `V` of the graph `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`.