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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 is said to be an out-neighbour of v. 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.
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
Digraphoccurs 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.
Digraph = digraph()Returns true if and only if the digraph
Digraph is acyclic.
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.
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 | {error, Reason}
Digraph = Subgraph = digraph()Options = [{type, SubgraphType}, {keep_labels, bool()}]Reason = {invalid_option, term()} | {unknown_type, term()}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, []).
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.
digraph(3)