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<erlref>
  <header>
    <copyright>
      <year>1996</year><year>2016</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.xml</file>
  </header>
  <module>digraph</module>
  <modulesummary>Directed graphs.</modulesummary>
  <description>
    <p>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.</p>

    <list type="bulleted">
      <item>
        <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).</p>
        <p>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>
      </item>
      <item>
        <p>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 <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>
      </item>
      <item>
        <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.</p>
      </item>
      <item>
        <p>The <marker id="out_degree"></marker><em>out-degree</em> of a vertex
          is the number of edges emanating from that vertex.</p>
      </item>
      <item>
        <p>The <marker id="in_degree"></marker><em>in-degree</em> of a vertex 
          is the number of edges incident on that vertex.</p>
      </item>
      <item>
        <p>If an edge is emanating from v and incident on w, then w is
          said to be an <marker id="out_neighbour"></marker>
          <em>out-neighbor</em> of v, and v is said to be an
          <marker id="in_neighbour"></marker><em>in-neighbor</em> of w.</p>
      </item>
      <item>
        <p>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.</p>
      </item>
      <item>
        <p>The <marker id="length"></marker><em>length</em> of path P is
          k-1.</p>
      </item>
      <item>
        <p>Path P is <marker id="simple_path"></marker><em>simple</em> if all
          vertices are distinct, except that the first and the last vertices
          can be the same.</p>
      </item>
      <item>
        <p>Path P is a <marker id="cycle"></marker><em>cycle</em> if the
          length of P is not zero and v[1] = v[k].</p>
      </item>
      <item>
        <p>A <marker id="loop"></marker><em>loop</em> is a cycle of length
          one.</p>
      </item>
      <item>
        <p>A <marker id="simple_cycle"></marker><em>simple cycle</em> is a path
          that is both a cycle and simple.</p>
      </item>
      <item>
        <p>An <marker id="acyclic_digraph"></marker><em>acyclic digraph</em>
          is a digraph without cycles.</p>
      </item>
    </list>
  </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
        <seealso marker="#new/0"><c>new/0,1</c></seealso>.</p></desc>
    </datatype>
    <datatype>
      <name>edge()</name>
    </datatype>
    <datatype>
      <name name="label"/>
    </datatype>
    <datatype>
      <name>vertex()</name>
    </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) edge <c><anno>E</anno></c>
          of 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 term <c>['$e'&nbsp;|&nbsp;N]</c>, where <c>N</c>
          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>,
          <c>{error,&nbsp;{bad_edge,&nbsp;<anno>Path</anno>}}</c> is returned.
          If <c><anno>G</anno></c> already has an edge with value
          <c><anno>E</anno></c> connecting a different pair of vertices,
          <c>{error,&nbsp;{bad_edge,&nbsp;[<anno>V1</anno>,&nbsp;<anno>V2</anno>]}}</c>
          is returned.
          If either of <c><anno>V1</anno></c> or <c><anno>V2</anno></c> is not
          a vertex of digraph <c><anno>G</anno></c>,
          <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) vertex
          <c><anno>V</anno></c> of 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 term <c>['$v'&nbsp;|&nbsp;N]</c>,
          where <c>N</c> 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 edge <c><anno>E</anno></c> from 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 list <c><anno>Edges</anno></c> from 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 digraph <c><anno>G</anno></c> until there are no
          <seealso marker="#path">paths</seealso> from vertex
          <c><anno>V1</anno></c> to vertex <c><anno>V2</anno></c>.</p>
        <p>A sketch of the procedure employed:</p>
        <list type="bulleted">
          <item>
            <p>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>.</p>
          </item>
          <item>
            <p>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).</p>
          </item>
          <item>
            <p>Repeat until there is no path between <c><anno>V1</anno></c>
              and <c><anno>V2</anno></c>.</p>
          </item>
	</list>
      </desc>
    </func>

    <func>
      <name name="del_vertex" arity="2"/>
      <fsummary>Delete a vertex from a digraph.</fsummary>
      <desc>
        <p>Deletes vertex <c><anno>V</anno></c> from 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 list <c><anno>Vertices</anno></c> from
          digraph <c><anno>G</anno></c>.</p>
      </desc>
    </func>

    <func>
      <name name="delete" arity="1"/>
      <fsummary>Delete a digraph.</fsummary>
      <desc>
        <p>Deletes digraph <c><anno>G</anno></c>. 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.</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 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 digraph <c><anno>G</anno></c>.
	  If no edge <c><anno>E</anno></c> of
          digraph <c><anno>G</anno></c> exists, <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 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 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 a <seealso marker="#simple_cycle">simple cycle</seealso> of
          length two or more exists through vertex <c><anno>V</anno></c>, the
          cycle is returned as a list
          <c>[<anno>V</anno>,&nbsp;...,&nbsp;<anno>V</anno>]</c> of vertices.
          If a <seealso marker="#loop">loop</seealso> through
          <c><anno>V</anno></c> exists, the loop is returned as a list
          <c>[<anno>V</anno>]</c>. If no cycles through
          <c><anno>V</anno></c> exist, <c>false</c> is returned.</p>
        <p><seealso marker="#get_path/3"><c>get_path/3</c></seealso> 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 vertex 
          <c><anno>V1</anno></c> to vertex <c><anno>V2</anno></c> of 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>Digraph <c><anno>G</anno></c> is traversed in a depth-first manner,
          and the first found path 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
          vertex <c><anno>V</anno></c> of 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.
          Notice that a <seealso marker="#loop">loop</seealso> through
          <c><anno>V</anno></c> is returned as list
          <c>[<anno>V</anno>,&nbsp;<anno>V</anno>]</c>.</p>
        <p><seealso marker="#get_short_path/3"><c>get_short_path/3</c></seealso>
          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 vertex
          <c><anno>V1</anno></c> to vertex <c><anno>V2</anno></c> of 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>Digraph <c><anno>G</anno></c> is traversed in a breadth-first
          manner, and the first found path 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
          vertex <c><anno>V</anno></c> of 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 digraph <c><anno>G</anno></c>,
          in some unspecified order.</p>
      </desc>
    </func>

    <func>
      <name name="in_neighbours" arity="2"/>
      <fsummary>Return all in-neighbors of a vertex of  a digraph.</fsummary>
      <desc>
        <p>Returns a list of
          all <seealso marker="#in_neighbour">in-neighbors</seealso> of 
          <c><anno>V</anno></c> of 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
          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 ETS 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><p>Allows <seealso marker="#cycle">cycles</seealso> in the
            digraph (default).</p></item>
          <tag><c>acyclic</c></tag>
          <item><p>The digraph is to be kept
          <seealso marker="#acyclic_digraph">acyclic</seealso>.</p></item>
          <tag><c>protected</c></tag>
          <item><p>Other processes can read the digraph (default).</p></item>
          <tag><c>private</c></tag>
          <item><p>The digraph can be read and modified by the creating
            process only.</p></item>
        </taglist>
        <p>If an unrecognized type option <c>T</c> is specified or
           <c><anno>Type</anno></c>
           is not a proper list, a <c>badarg</c> exception is raised.</p>
      </desc>
    </func>

    <func>
      <name name="no_edges" arity="1"/>
      <fsummary>Return the number of edges of a digraph.</fsummary>
      <desc>
        <p>Returns the number of edges of 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 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
          vertex <c><anno>V</anno></c> of 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 digraph <c><anno>G</anno></c>,
          in some unspecified order.</p>
      </desc>
    </func>

    <func>
      <name name="out_neighbours" arity="2"/>
      <fsummary>Return all out-neighbors of a vertex of a digraph.</fsummary>
      <desc>
        <p>Returns a list of 
          all <seealso marker="#out_neighbour">out-neighbors</seealso> of 
          <c><anno>V</anno></c> of 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 digraph <c><anno>G</anno></c>,
          or <c>false</c> if no vertex <c><anno>V</anno></c>
          of digraph <c><anno>G</anno></c> exists.</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 digraph <c><anno>G</anno></c>, in
          some unspecified order.</p>
      </desc>
    </func>
  </funcs>

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

