# digraph_utils

## Algorithms for Directed Graphs

The `digraph_utils` module implements some algorithms based on depth-first traversal of directed graphs. See the `digraph` module for basic functions on directed graphs.

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).

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.

An edge e = (v, w) is said to emanate from vertex v and to be incident on vertex w. 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[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 the path P is k-1. 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. An acyclic digraph is a digraph that has no cycles.

A depth-first traversal of a directed digraph can be viewed as a process that visits all vertices of the digraph. Initially, all vertices are marked as unvisited. The traversal starts with an arbitrarily chosen vertex, which is marked as visited, and follows an edge to an unmarked vertex, marking that vertex. The search then proceeds from that vertex in the same fashion, until there is no edge leading to an unvisited vertex. At that point the process backtracks, and the traversal continues as long as there are unexamined edges. If there remain unvisited vertices when all edges from the first vertex have been examined, some hitherto unvisited vertex is chosen, and the process is repeated.

A partial ordering of a set S is a transitive, antisymmetric and reflexive relation between the objects of S. The problem of topological sorting is to find a total ordering of S that is a superset of the partial ordering. A digraph G = (V, E) is equivalent to a relation E on V (we neglect the fact that the version of directed graphs implemented in the `digraph` module allows multiple edges between vertices). If the digraph has no cycles of length two or more, then the reflexive and transitive closure of E is a partial ordering.

A subgraph G' of G is a digraph whose vertices and edges form subsets of the vertices and edges of G. G' is maximal with respect to a property P if all other subgraphs that include the vertices of G' do not have the property P. A strongly connected component is a maximal subgraph such that there is a path between each pair of vertices. A connected component is a maximal subgraph such that there is a path between each pair of vertices, considering all edges undirected. An arborescence is an acyclic digraph with a vertex V, the root, such that there is a unique path from V to every other vertex of G. A tree is an acyclic non-empty digraph such that there is a unique path between every pair of vertices, considering all edges undirected.

### arborescence_root(Digraph) -> no | {yes, Root}

• `Digraph = digraph()`
• `Root = vertex()`

Returns `{yes, Root}` if `Root` is the root of the arborescence `Digraph`, `no` otherwise.

### components(Digraph) -> [Component]

• `Digraph = digraph()`
• `Component = [vertex()]`

Returns a list of connected components. Each component is represented by its vertices. The order of the vertices and the order of the components are arbitrary. Each vertex of the digraph `Digraph` occurs in exactly one component.

### condensation(Digraph) -> CondensedDigraph

• `Digraph = CondensedDigraph = digraph()`

Creates a digraph where the vertices are the strongly connected components of `Digraph` as returned by `strong_components/1`. If X and Y are strongly connected components, and there exist vertices x and y in X and Y respectively such that there is an edge emanating from x and incident on y, then an edge emanating from X and incident on Y is created.

The created digraph has the same type as `Digraph`. All vertices and edges have the default label `[]`.

Each and every cycle is included in some strongly connected component, which implies that there always exists a topological ordering of the created digraph.

### cyclic_strong_components(Digraph) -> [StrongComponent]

• `Digraph = digraph()`
• `StrongComponent = [vertex()]`

Returns a list of strongly connected components. Each strongly component is represented by its vertices. The order of the vertices and the order of the components are arbitrary. Only vertices that are included in some cycle in `Digraph` are returned, otherwise the returned list is equal to that returned by `strong_components/1`.

### is_acyclic(Digraph) -> bool()

• `Digraph = digraph()`

Returns `true` if and only if the digraph `Digraph` is acyclic.

### is_arborescence(Digraph) -> bool()

• `Digraph = digraph()`

Returns `true` if and only if the digraph `Digraph` is an arborescence.

### is_tree(Digraph) -> bool()

• `Digraph = digraph()`

Returns `true` if and only if the digraph `Digraph` is a tree.

### loop_vertices(Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns a list of all vertices of `Digraph` that are included in some loop.

### postorder(Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns all vertices of the digraph `Digraph`. The order is given by a depth-first traversal of the digraph, collecting visited vertices in postorder. More precisely, the vertices visited while searching from an arbitrarily chosen vertex are collected in postorder, and all those collected vertices are placed before the subsequently visited vertices.

### preorder(Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns all vertices of the digraph `Digraph`. The order is given by a depth-first traversal of the digraph, collecting visited vertices in pre-order.

### reachable(Vertices, Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns an unsorted list of digraph vertices such that for each vertex in the list, there is a path in `Digraph` from some vertex of `Vertices` to the vertex. In particular, since paths may have length zero, the vertices of `Vertices` are included in the returned list.

### reachable_neighbours(Vertices, Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns an unsorted list of digraph vertices such that for each vertex in the list, there is a path in `Digraph` of length one or more from some vertex of `Vertices` to the vertex. As a consequence, only those vertices of `Vertices` that are included in some cycle are returned.

### reaching(Vertices, Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns an unsorted list of digraph vertices such that for each vertex in the list, there is a path from the vertex to some vertex of `Vertices`. In particular, since paths may have length zero, the vertices of `Vertices` are included in the returned list.

### reaching_neighbours(Vertices, Digraph) -> Vertices

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns an unsorted list of digraph vertices such that for each vertex in the list, there is a path of length one or more from the vertex to some vertex of `Vertices`. As a consequence, only those vertices of `Vertices` that are included in some cycle are returned.

### strong_components(Digraph) -> [StrongComponent]

• `Digraph = digraph()`
• `StrongComponent = [vertex()]`

Returns a list of strongly connected components. Each strongly component is represented by its vertices. The order of the vertices and the order of the components are arbitrary. Each vertex of the digraph `Digraph` occurs in exactly one strong component.

### subgraph(Digraph, Vertices [, Options]) -> Subgraph

• `Digraph = Subgraph = digraph()`
• `Options = [{type, SubgraphType}, {keep_labels, bool()}]`
• `SubgraphType = inherit | type()`
• `Vertices = [vertex()]`

Creates a maximal subgraph of `Digraph` having as vertices those vertices of `Digraph` that are mentioned in `Vertices`.

If the value of the option `type` is `inherit`, which is the default, then the type of `Digraph` is used for the subgraph as well. Otherwise the option value of `type` is used as argument to `digraph:new/1`.

If the value of the option `keep_labels` is `true`, which is the default, then the labels of vertices and edges of `Digraph` are used for the subgraph as well. If the value is `false`, then the default label, `[]`, is used for the subgraph's vertices and edges.

`subgraph(Digraph, Vertices)` is equivalent to `subgraph(Digraph, Vertices, [])`.

There will be a `badarg` exception if any of the arguments are invalid.

### topsort(Digraph) -> Vertices | false

• `Digraph = digraph()`
• `Vertices = [vertex()]`

Returns a topological ordering of the vertices of the digraph `Digraph` if such an ordering exists, `false` otherwise. For each vertex in the returned list, there are no out-neighbours that occur earlier in the list.