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NAME

       digraph - Directed graphs.

DESCRIPTION

       This  module provides a version of labeled directed graphs. What makes the graphs provided
       here non-proper directed graphs is that  multiple  edges  between  vertices  are  allowed.
       However, the customary definition of directed graphs is used here.

         * 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 x 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 more information. Such information can be  attached  to
           the vertices and to the edges of the digraph. An annotated digraph 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 an edge is emanating from v and incident on w, then w is said to be an out-neighbor
           of v, and v is said to be an in-neighbor 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 path P is k-1.

         * Path P is simple if all vertices are distinct, except that  the  first  and  the  last
           vertices can be the same.

         * Path 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.

         * A simple cycle is a path that is both a cycle and simple.

         * An acyclic digraph is a digraph without cycles.

DATA TYPES

       d_type() = d_cyclicity() | d_protection()

       d_cyclicity() = acyclic | cyclic

       d_protection() = private | protected

       graph()

              A digraph as returned by new/0,1.

       edge()

       label() = term()

       vertex()

EXPORTS

       add_edge(G, V1, V2) -> edge() | {error, add_edge_err_rsn()}

       add_edge(G, V1, V2, Label) -> edge() | {error, add_edge_err_rsn()}

       add_edge(G, E, V1, V2, Label) ->
                   edge() | {error, add_edge_err_rsn()}

              Types:

                 G = graph()
                 E = edge()
                 V1 = V2 = vertex()
                 Label = label()
                 add_edge_err_rsn() =
                     {bad_edge, Path :: [vertex()]} | {bad_vertex, V :: vertex()}

              add_edge/5  creates  (or  modifies)  edge  E of 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 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, {error,  {bad_edge,  Path}}
              is  returned.  If G already has an edge with value E connecting a different pair of
              vertices, {error, {bad_edge, [V1, V2]}} is returned. If either of V1 or V2 is not a
              vertex of digraph G, {error, {bad_vertex, V}} is returned, V = V1 or V = V2.

       add_vertex(G) -> vertex()

       add_vertex(G, V) -> vertex()

       add_vertex(G, V, Label) -> vertex()

              Types:

                 G = graph()
                 V = vertex()
                 Label = label()

              add_vertex/3  creates (or modifies) vertex V of 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 term ['$v' | N], where N is an
              integer >= 0.

       del_edge(G, E) -> true

              Types:

                 G = graph()
                 E = edge()

              Deletes edge E from digraph G.

       del_edges(G, Edges) -> true

              Types:

                 G = graph()
                 Edges = [edge()]

              Deletes the edges in list Edges from digraph G.

       del_path(G, V1, V2) -> true

              Types:

                 G = graph()
                 V1 = V2 = vertex()

              Deletes edges from digraph G until there are no paths from vertex V1 to vertex V2.

              A sketch of the procedure employed:

                * Find an arbitrary simple path v[1], v[2], ..., 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

              Types:

                 G = graph()
                 V = vertex()

              Deletes  vertex  V  from digraph G. Any edges emanating from V or incident on V are
              also deleted.

       del_vertices(G, Vertices) -> true

              Types:

                 G = graph()
                 Vertices = [vertex()]

              Deletes the vertices in list Vertices from digraph G.

       delete(G) -> true

              Types:

                 G = graph()

              Deletes digraph G. This call is important as digraphs  are  implemented  with  ETS.
              There  is  no  garbage collection of ETS tables. However, the digraph is deleted if
              the process that created the digraph terminates.

       edge(G, E) -> {E, V1, V2, Label} | false

              Types:

                 G = graph()
                 E = edge()
                 V1 = V2 = vertex()
                 Label = label()

              Returns {E, V1, V2, Label}, where Label is the label of edge E  emanating  from  V1
              and  incident  on  V2  of  digraph  G.  If  no edge E of digraph G exists, false is
              returned.

       edges(G) -> Edges

              Types:

                 G = graph()
                 Edges = [edge()]

              Returns a list of all edges of digraph G, in some unspecified order.

       edges(G, V) -> Edges

              Types:

                 G = graph()
                 V = vertex()
                 Edges = [edge()]

              Returns a list of all edges emanating from or incident onV of digraph  G,  in  some
              unspecified order.

       get_cycle(G, V) -> Vertices | false

              Types:

                 G = graph()
                 V = vertex()
                 Vertices = [vertex(), ...]

              If  a  simple  cycle  of  length  two or more exists through vertex V, the cycle is
              returned as a list [V, ..., V] of vertices. If a loop through V exists, the loop is
              returned as a list [V]. If no cycles through V exist, false is returned.

              get_path/3 is used for finding a simple cycle through V.

       get_path(G, V1, V2) -> Vertices | false

              Types:

                 G = graph()
                 V1 = V2 = vertex()
                 Vertices = [vertex(), ...]

              Tries  to  find a simple path from vertex V1 to vertex V2 of 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.

              Digraph  G  is  traversed  in  a  depth-first  manner,  and the first found path is
              returned.

       get_short_cycle(G, V) -> Vertices | false

              Types:

                 G = graph()
                 V = vertex()
                 Vertices = [vertex(), ...]

              Tries to find an as short as possible simple cycle through vertex V of  digraph  G.
              Returns  the  cycle  as a list [V, ..., V] of vertices, or false if no simple cycle
              through V exists. Notice that a loop through V is returned as list [V, V].

              get_short_path/3 is used for finding a simple cycle through V.

       get_short_path(G, V1, V2) -> Vertices | false

              Types:

                 G = graph()
                 V1 = V2 = vertex()
                 Vertices = [vertex(), ...]

              Tries to find an as short as possible simple path from vertex V1 to  vertex  V2  of
              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.

              Digraph G is traversed in a breadth-first manner,  and  the  first  found  path  is
              returned.

       in_degree(G, V) -> integer() >= 0

              Types:

                 G = graph()
                 V = vertex()

              Returns the in-degree of vertex V of digraph G.

       in_edges(G, V) -> Edges

              Types:

                 G = graph()
                 V = vertex()
                 Edges = [edge()]

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

       in_neighbours(G, V) -> Vertex

              Types:

                 G = graph()
                 V = vertex()
                 Vertex = [vertex()]

              Returns a list of all in-neighbors of V of digraph G, in some unspecified order.

       info(G) -> InfoList

              Types:

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

              Returns  a list of {Tag, Value} pairs describing 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() -> graph()

              Equivalent to new([]).

       new(Type) -> graph()

              Types:

                 Type = [d_type()]
                 d_type() = d_cyclicity() | d_protection()
                 d_cyclicity() = acyclic | cyclic
                 d_protection() = private | protected

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

                cyclic:
                  Allows 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 specified or Type is not a proper list, a
              badarg exception is raised.

       no_edges(G) -> integer() >= 0

              Types:

                 G = graph()

              Returns the number of edges of digraph G.

       no_vertices(G) -> integer() >= 0

              Types:

                 G = graph()

              Returns the number of vertices of digraph G.

       out_degree(G, V) -> integer() >= 0

              Types:

                 G = graph()
                 V = vertex()

              Returns the out-degree of vertex V of digraph G.

       out_edges(G, V) -> Edges

              Types:

                 G = graph()
                 V = vertex()
                 Edges = [edge()]

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

       out_neighbours(G, V) -> Vertices

              Types:

                 G = graph()
                 V = vertex()
                 Vertices = [vertex()]

              Returns a list of all out-neighbors of V of digraph G, in some unspecified order.

       vertex(G, V) -> {V, Label} | false

              Types:

                 G = graph()
                 V = vertex()
                 Label = label()

              Returns {V, Label}, where Label is the label of the vertex V of digraph G, or false
              if no vertex V of digraph G exists.

       vertices(G) -> Vertices

              Types:

                 G = graph()
                 Vertices = [vertex()]

              Returns a list of all vertices of digraph G, in some unspecified order.

SEE ALSO

       digraph_utils(3erl), ets(3erl)