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}

Types:

Digraph = digraph()

Root = vertex()

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

components(Digraph) -> [Component]

Types:

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

Types:

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]

Types:

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.

Types:

Digraph = digraph()

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

is_arborescence(Digraph) -> bool()

Types:

Digraph = digraph()

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

Types:

Digraph = digraph()

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

loop_vertices(Digraph) -> Vertices

Types:

Digraph = digraph()

Vertices = [vertex()]

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

postorder(Digraph) -> Vertices

Types:

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.

Types:

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

Types:

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

Types:

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

Types:

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

Types:

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]

Types:

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

Types:

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

Types:

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.

stdlib 1.16

Copyright © 1991-2009 Ericsson AB