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<erlref>
  <header>
    <copyright>
      <year>1996</year><year>2014</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      Licensed under the Apache License, Version 2.0 (the "License");
      you may not use this file except in compliance with the License.
      You may obtain a copy of the License at
 
          http://www.apache.org/licenses/LICENSE-2.0

      Unless required by applicable law or agreed to in writing, software
      distributed under the License is distributed on an "AS IS" BASIS,
      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
      See the License for the specific language governing permissions and
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    </legalnotice>

    <title>digraph</title>
    <prepared>Tony</prepared>
    <responsible>Bjarne D&auml;cker</responsible>
    <docno>1</docno>
    <approved>Bjarne D&auml;cker</approved>
    <checked></checked>
    <date>2001-08-27</date>
    <rev>C</rev>
    <file>digraph.sgml</file>
  </header>
  <module>digraph</module>
  <modulesummary>Directed Graphs</modulesummary>
  <description>
    <p>The <c>digraph</c> 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.
      </p>
    <p>A <marker id="digraph"></marker><em>directed graph</em> (or just
      "digraph") is a pair (V,&nbsp;E) of a finite set V of 
      <marker id="vertex"></marker><em>vertices</em> and a finite set E of
      <marker id="edge"></marker><em>directed edges</em> (or just "edges"). 
      The set of
      edges E is a subset of V&nbsp;&times;&nbsp;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 
      <marker id="empty_digraph"></marker><em>empty digraph</em>. 
      Both vertices and edges are represented by unique Erlang terms.
      </p>
    <p>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
      <em>labeled digraph</em>, and the information attached to a
      vertex or an edge is called a <marker id="label"></marker>
      <em>label</em>. Labels are Erlang terms.
      </p>
    <p>An edge e&nbsp;=&nbsp;(v,&nbsp;w) is said to 
      <marker id="emanate"></marker><em>emanate</em> from vertex v and 
      to be <marker id="incident"></marker><em>incident</em> on vertex w. 
      The <marker id="out_degree"></marker><em>out-degree</em> of a vertex 
      is the number of edges emanating from that vertex. 
      The <marker id="in_degree"></marker><em>in-degree</em> 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 
      said to be an <marker id="out_neighbour"></marker>
      <em>out-neighbour</em> of v, and v is said to be an 
      <marker id="in_neighbour"></marker><em>in-neighbour</em> of w. 
      A <marker id="path"></marker><em>path</em> P from v[1] to v[k] 
      in a digraph (V,&nbsp;E) is a non-empty sequence
      v[1],&nbsp;v[2],&nbsp;...,&nbsp;v[k] of vertices in V such that
      there is an edge (v[i],v[i+1]) in E for
      1&nbsp;&lt;=&nbsp;i&nbsp;&lt;&nbsp;k. 
      The <marker id="length"></marker><em>length</em> of the path P is k-1. 
      P is <marker id="simple_path"></marker><em>simple</em> if all 
      vertices are distinct, except that the first and the last vertices
      may be the same. 
      P is a <marker id="cycle"></marker><em>cycle</em> if the length 
      of P is not zero and v[1] = v[k]. 
      A <marker id="loop"></marker><em>loop</em> is a cycle of length one. 
      A <marker id="simple_cycle"></marker><em>simple cycle</em> is a path
      that is both a cycle and simple. 
      An <marker id="acyclic_digraph"></marker><em>acyclic digraph</em>
      is a digraph that has no cycles.
      </p>
  </description>
  <datatypes>
    <datatype>
      <name name="d_type"/>
    </datatype>
    <datatype>
      <name name="d_cyclicity"/>
    </datatype>
    <datatype>
      <name name="d_protection"/>
    </datatype>
    <datatype>
      <name name="graph"/>
      <desc><p>A digraph as returned by <c>new/0,1</c>.</p></desc>
    </datatype>
    <datatype>
      <name>edge()</name>
      <desc><p><marker id="type-edge"/></p></desc>
    </datatype>
    <datatype>
      <name name="label"/>
    </datatype>
    <datatype>
      <name>vertex()</name>
      <desc><p><marker id="type-vertex"/></p></desc>
    </datatype>
  </datatypes>
  <funcs>
    <func>
      <name name="add_edge" arity="3"/>
      <name name="add_edge" arity="4"/>
      <name name="add_edge" arity="5"/>
      <fsummary>Add an edge to a digraph.</fsummary>
      <type name="add_edge_err_rsn"/>
      <desc>
        <p><c>add_edge/5</c> creates (or modifies) the edge <c><anno>E</anno></c>
          of the digraph <c><anno>G</anno></c>, using <c><anno>Label</anno></c> as the (new)
          <seealso marker="#label">label</seealso> of the edge. The
          edge is <seealso marker="#emanate">emanating</seealso> from
          <c><anno>V1</anno></c> and <seealso marker="#incident">incident</seealso>
          on <c><anno>V2</anno></c>. Returns <c><anno>E</anno></c>.
          </p>
        <p><c>add_edge(<anno>G</anno>,&nbsp;<anno>V1</anno>,&nbsp;<anno>V2</anno>,&nbsp;<anno>Label</anno>)</c> is
          equivalent to
          <c>add_edge(<anno>G</anno>,&nbsp;<anno>E</anno>,&nbsp;<anno>V1</anno>,&nbsp;<anno>V2</anno>,&nbsp;<anno>Label</anno>)</c>,
          where <c><anno>E</anno></c> is a created edge. The created edge is
          represented by the term <c>['$e'&nbsp;|&nbsp;N]</c>, where N
          is an integer&nbsp;&gt;=&nbsp;0.
          </p>
        <p><c>add_edge(<anno>G</anno>,&nbsp;<anno>V1</anno>,&nbsp;<anno>V2</anno>)</c> is equivalent to
          <c>add_edge(<anno>G</anno>,&nbsp;<anno>V1</anno>,&nbsp;<anno>V2</anno>,&nbsp;[])</c>.
          </p>
        <p>If the edge would create a cycle in 
	  an <seealso marker="#acyclic_digraph">acyclic digraph</seealso>,
	  then <c>{error,&nbsp;{bad_edge,&nbsp;<anno>Path</anno>}}</c> is returned. If
          either of <c><anno>V1</anno></c> or <c><anno>V2</anno></c> is not a vertex of the
          digraph <c><anno>G</anno></c>, then
          <c>{error,&nbsp;{bad_vertex,&nbsp;</c><anno>V</anno><c>}}</c> is
          returned, <anno>V</anno>&nbsp;=&nbsp;<c><anno>V1</anno></c> or
          <anno>V</anno>&nbsp;=&nbsp;<c><anno>V2</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="add_vertex" arity="1"/>
      <name name="add_vertex" arity="2"/>
      <name name="add_vertex" arity="3"/>
      <fsummary>Add or modify a vertex of a digraph.</fsummary>
      <desc>
        <p><c>add_vertex/3</c> creates (or modifies) the vertex <c><anno>V</anno></c>
          of the digraph <c><anno>G</anno></c>, using <c><anno>Label</anno></c> as the (new)
          <seealso marker="#label">label</seealso> of the
          vertex. Returns <c><anno>V</anno></c>.
          </p>
        <p><c>add_vertex(<anno>G</anno>,&nbsp;<anno>V</anno>)</c> is equivalent to
          <c>add_vertex(<anno>G</anno>,&nbsp;<anno>V</anno>,&nbsp;[])</c>.
          </p>
        <p><c>add_vertex/1</c> creates a vertex using the empty list
          as label, and returns the created vertex. The created vertex
          is represented by the term <c>['$v'&nbsp;|&nbsp;N]</c>,
          where N is an integer&nbsp;&gt;=&nbsp;0.
          </p>
      </desc>
    </func>
    <func>
      <name name="del_edge" arity="2"/>
      <fsummary>Delete an edge from a digraph.</fsummary>
      <desc>
        <p>Deletes the edge <c><anno>E</anno></c> from the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="del_edges" arity="2"/>
      <fsummary>Delete edges from a digraph.</fsummary>
      <desc>
        <p>Deletes the edges in the list <c><anno>Edges</anno></c> from the digraph
          <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="del_path" arity="3"/>
      <fsummary>Delete paths from a digraph.</fsummary>
      <desc>
        <p>Deletes edges from the digraph <c><anno>G</anno></c> until there are no
          <seealso marker="#path">paths</seealso> from the vertex
          <c><anno>V1</anno></c> to the vertex <c><anno>V2</anno></c>.
          </p>
        <p>A sketch of the procedure employed: Find an arbitrary
          <seealso marker="#simple_path">simple path</seealso>
          v[1],&nbsp;v[2],&nbsp;...,&nbsp;v[k] from <c><anno>V1</anno></c> to
          <c><anno>V2</anno></c> in <c><anno>G</anno></c>. Remove all edges of
	  <c><anno>G</anno></c> <seealso marker="#emanate">emanating</seealso> from v[i]
	  and <seealso marker="#incident">incident</seealso> to v[i+1] for
          1&nbsp;&lt;=&nbsp;i&nbsp;&lt;&nbsp;k (including multiple
          edges). Repeat until there is no path between <c><anno>V1</anno></c> and
          <c><anno>V2</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="del_vertex" arity="2"/>
      <fsummary>Delete a vertex from a digraph.</fsummary>
      <desc>
        <p>Deletes the vertex <c><anno>V</anno></c> from the digraph <c><anno>G</anno></c>. Any
          edges <seealso marker="#emanate">emanating</seealso> from
          <c><anno>V</anno></c> or <seealso marker="#incident">incident</seealso>
          on <c><anno>V</anno></c> are also deleted.
          </p>
      </desc>
    </func>
    <func>
      <name name="del_vertices" arity="2"/>
      <fsummary>Delete vertices from a digraph.</fsummary>
      <desc>
        <p>Deletes the vertices in the list <c><anno>Vertices</anno></c> from the
          digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="delete" arity="1"/>
      <fsummary>Delete a digraph.</fsummary>
      <desc>
        <p>Deletes the digraph <c><anno>G</anno></c>. This call is important
          because digraphs are implemented with <c>ETS</c>. There is
          no garbage collection of <c>ETS</c> tables. The digraph
          will, however, be deleted if the process that created the
          digraph terminates.
          </p>
      </desc>
    </func>
    <func>
      <name name="edge" arity="2"/>
      <fsummary>Return the vertices and the label of an edge  of a digraph.</fsummary>
      <desc>
        <p>Returns <c>{<anno>E</anno>,&nbsp;<anno>V1</anno>,&nbsp;<anno>V2</anno>,&nbsp;<anno>Label</anno>}</c> where
          <c><anno>Label</anno></c> is the <seealso marker="#label">label</seealso>
          of the edge 
	  <c><anno>E</anno></c> <seealso marker="#emanate">emanating</seealso> from
	  <c><anno>V1</anno></c> and <seealso marker="#incident">incident</seealso> on
	  <c><anno>V2</anno></c> of the digraph <c><anno>G</anno></c>.
	  If there is no edge <c><anno>E</anno></c> of the
          digraph <c><anno>G</anno></c>, then <c>false</c> is returned.
          </p>
      </desc>
    </func>
    <func>
      <name name="edges" arity="1"/>
      <fsummary>Return all edges of a digraph.</fsummary>
      <desc>
        <p>Returns a list of all edges of the digraph <c><anno>G</anno></c>, in
          some unspecified order.
          </p>
      </desc>
    </func>
    <func>
      <name name="edges" arity="2"/>
      <fsummary>Return the edges emanating from or incident on  a vertex of a digraph.</fsummary>
      <desc>
        <p>Returns a list of all 
	edges <seealso marker="#emanate">emanating</seealso> from 
	or <seealso marker="#incident">incident</seealso> on <c><anno>V</anno></c>
	of the digraph <c><anno>G</anno></c>, in some unspecified order.</p>
      </desc>
    </func>
    <func>
      <name name="get_cycle" arity="2"/>
      <fsummary>Find one cycle in a digraph.</fsummary>
      <desc>
        <p>If there is 
	  a <seealso marker="#simple_cycle">simple cycle</seealso> of 
	  length two or more through the vertex
          <c><anno>V</anno></c>, then the cycle is returned as a list
          <c>[<anno>V</anno>,&nbsp;...,&nbsp;<anno>V</anno>]</c> of vertices, otherwise if there
          is a <seealso marker="#loop">loop</seealso> through
          <c><anno>V</anno></c>, then the loop is returned as a list <c>[<anno>V</anno>]</c>. If
          there are no cycles through <c><anno>V</anno></c>, then <c>false</c> is
          returned.
          </p>
        <p><c>get_path/3</c> is used for finding a simple cycle
          through <c><anno>V</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="get_path" arity="3"/>
      <fsummary>Find one path in a digraph.</fsummary>
      <desc>
        <p>Tries to find 
	  a <seealso marker="#simple_path">simple path</seealso> from 
	  the vertex <c><anno>V1</anno></c> to the vertex
          <c><anno>V2</anno></c> of the digraph <c><anno>G</anno></c>. Returns the path as a
          list <c>[<anno>V1</anno>,&nbsp;...,&nbsp;<anno>V2</anno>]</c> of vertices, or
          <c>false</c> if no simple path from <c><anno>V1</anno></c> to <c><anno>V2</anno></c>
          of length one or more exists.
          </p>
        <p>The digraph <c><anno>G</anno></c> is traversed in a depth-first manner,
          and the first path found is returned.
          </p>
      </desc>
    </func>
    <func>
      <name name="get_short_cycle" arity="2"/>
      <fsummary>Find one short cycle in a digraph.</fsummary>
      <desc>
        <p>Tries to find an as short as 
	  possible <seealso marker="#simple_cycle">simple cycle</seealso> through
	  the vertex <c><anno>V</anno></c> of the digraph <c>G</c>. Returns the cycle
          as a list <c>[<anno>V</anno>,&nbsp;...,&nbsp;<anno>V</anno>]</c> of vertices, or
          <c>false</c> if no simple cycle through <c><anno>V</anno></c> exists.
          Note that a <seealso marker="#loop">loop</seealso> through
          <c><anno>V</anno></c> is returned as the list <c>[<anno>V</anno>,&nbsp;<anno>V</anno>]</c>.
          </p>
        <p><c>get_short_path/3</c> is used for finding a simple cycle
          through <c><anno>V</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="get_short_path" arity="3"/>
      <fsummary>Find one short path in a digraph.</fsummary>
      <desc>
        <p>Tries to find an as short as 
	  possible <seealso marker="#simple_path">simple path</seealso> from
	  the vertex <c><anno>V1</anno></c> to the vertex <c><anno>V2</anno></c> of the digraph <c><anno>G</anno></c>.
          Returns the path as a list <c>[<anno>V1</anno>,&nbsp;...,&nbsp;<anno>V2</anno>]</c> of
          vertices, or <c>false</c> if no simple path from <c><anno>V1</anno></c>
          to <c><anno>V2</anno></c> of length one or more exists.
          </p>
        <p>The digraph <c><anno>G</anno></c> is traversed in a breadth-first
          manner, and the first path found is returned.
          </p>
      </desc>
    </func>
    <func>
      <name name="in_degree" arity="2"/>
      <fsummary>Return the in-degree of a vertex of a digraph.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#in_degree">in-degree</seealso> of the vertex
          <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="in_edges" arity="2"/>
      <fsummary>Return all edges incident on a vertex of  a digraph.</fsummary>
      <desc>
        <p>Returns a list of all 
	  edges <seealso marker="#incident">incident</seealso> on
	  <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>, in some unspecified order.
          </p>
      </desc>
    </func>
    <func>
      <name name="in_neighbours" arity="2"/>
      <fsummary>Return all in-neighbours of a vertex of  a digraph.</fsummary>
      <desc>
        <p>Returns a list of 
	  all <seealso marker="#in_neighbour">in-neighbours</seealso> of 
	  <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>, in some unspecified order.
          </p>
      </desc>
    </func>
    <func>
      <name name="info" arity="1"/>
      <fsummary>Return information about a digraph.</fsummary>
      <type name="d_cyclicity"/>
      <type name="d_protection"/>
      <desc>
        <p>Returns a list of <c>{Tag, Value}</c> pairs describing the
          digraph <c><anno>G</anno></c>. The following pairs are returned:
          </p>
        <list type="bulleted">
          <item>
            <p><c>{cyclicity, <anno>Cyclicity</anno>}</c>, where <c><anno>Cyclicity</anno></c>
              is <c>cyclic</c> or <c>acyclic</c>, according to the
              options given to <c>new</c>.</p>
          </item>
          <item>
            <p><c>{memory, <anno>NoWords</anno>}</c>, where <c><anno>NoWords</anno></c> is
              the number of words allocated to the <c>ETS</c> tables.</p>
          </item>
          <item>
            <p><c>{protection, <anno>Protection</anno>}</c>, where <c><anno>Protection</anno></c>
              is <c>protected</c> or <c>private</c>, according
              to the options given to <c>new</c>.</p>
          </item>
        </list>
      </desc>
    </func>
    <func>
      <name name="new" arity="0"/>
      <fsummary>Return a protected empty digraph, where cycles  are allowed.</fsummary>
      <desc>
        <p>Equivalent to <c>new([])</c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="new" arity="1"/>
      <fsummary>Create a new empty digraph.</fsummary>
      <type variable="Type"/>
      <type name="d_type"/>
      <type name="d_cyclicity"/>
      <type name="d_protection"/>
      <desc>
        <p>Returns 
	  an <seealso marker="#empty_digraph">empty digraph</seealso> with 
	  properties according to the options in <c><anno>Type</anno></c>:</p>
        <taglist>
          <tag><c>cyclic</c></tag>
          <item>Allow <seealso marker="#cycle">cycles</seealso> in the
           digraph (default).</item>
          <tag><c>acyclic</c></tag>
          <item>The digraph is to be kept <seealso marker="#acyclic_digraph">acyclic</seealso>.</item>
          <tag><c>protected</c></tag>
          <item>Other processes can read the digraph (default).</item>
          <tag><c>private</c></tag>
          <item>The digraph can be read and modified by the creating
           process only.</item>
        </taglist>
        <p>If an unrecognized type option <c>T</c> is given or <c><anno>Type</anno></c>
	   is not a proper list, there will be a <c>badarg</c> exception.
          </p>
      </desc>
    </func>
    <func>
      <name name="no_edges" arity="1"/>
      <fsummary>Return the number of edges of the a digraph.</fsummary>
      <desc>
        <p>Returns the number of edges of the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="no_vertices" arity="1"/>
      <fsummary>Return the number of vertices of a digraph.</fsummary>
      <desc>
        <p>Returns the number of vertices of the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="out_degree" arity="2"/>
      <fsummary>Return the out-degree of a vertex of a digraph.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#out_degree">out-degree</seealso> of the vertex
          <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="out_edges" arity="2"/>
      <fsummary>Return all edges emanating from a vertex of  a digraph.</fsummary>
      <desc>
        <p>Returns a list of all 
	  edges <seealso marker="#emanate">emanating</seealso> from 
          <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>, in some unspecified order.
          </p>
      </desc>
    </func>
    <func>
      <name name="out_neighbours" arity="2"/>
      <fsummary>Return all out-neighbours of a vertex of  a digraph.</fsummary>
      <desc>
        <p>Returns a list of 
	  all <seealso marker="#out_neighbour">out-neighbours</seealso> of 
	  <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>, in some unspecified order.
          </p>
      </desc>
    </func>
    <func>
      <name name="vertex" arity="2"/>
      <fsummary>Return the label of a vertex of a digraph.</fsummary>
      <desc>
        <p>Returns <c>{<anno>V</anno>,&nbsp;<anno>Label</anno>}</c> where <c><anno>Label</anno></c> is the
          <seealso marker="#label">label</seealso> of the vertex
          <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>, or <c>false</c> if there
          is no vertex <c><anno>V</anno></c> of the digraph <c><anno>G</anno></c>.
          </p>
      </desc>
    </func>
    <func>
      <name name="vertices" arity="1"/>
      <fsummary>Return all vertices of a digraph.</fsummary>
      <desc>
        <p>Returns a list of all vertices of the digraph <c><anno>G</anno></c>, in
          some unspecified order.
          </p>
      </desc>
    </func>
  </funcs>

  <section>
    <title>See Also</title>
    <p><seealso marker="digraph_utils">digraph_utils(3)</seealso>, 
      <seealso marker="ets">ets(3)</seealso></p>
  </section>
</erlref>

