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  <header>
    <copyright>
      <year>2000</year><year>2021</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.
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          http://www.apache.org/licenses/LICENSE-2.0

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      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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    </legalnotice>

    <title>digraph_utils</title>
    <prepared>Hans Bolinder</prepared>
    <responsible></responsible>
    <docno></docno>
    <approved></approved>
    <checked></checked>
    <date>2001-08-27</date>
    <rev>PA1</rev>
    <file>digraph_utils.xml</file>
  </header>
  <module since="">digraph_utils</module>
  <modulesummary>Algorithms for directed graphs.</modulesummary>
  <description>
    <p>This module provides algorithms based on depth-first traversal of
      directed graphs. For basic functions on directed graphs, see the
      <seeerl marker="digraph"><c>digraph(3)</c></seeerl> module.</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>
      </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>.</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>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 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>An <marker id="acyclic_digraph"></marker><em>acyclic digraph</em>
          is a digraph without cycles.</p>
      </item>
      <item>
        <p>A <marker id="depth_first_traversal"></marker><em>depth-first
          traversal</em> 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 unvisited vertices remain
          when all edges from the first vertex have been examined, some
          so far unvisited vertex is chosen, and the process is repeated.</p>
      </item>
      <item>
        <p>A <marker id="partial_ordering"></marker><em>partial ordering</em>
          of a set S is a transitive, antisymmetric, and reflexive relation
          between the objects of S.</p>
      </item>
      <item>
        <p>The problem of
          <marker id="topsort"></marker><em>topological sorting</em> is to find
          a total ordering of S that is a superset of the partial ordering. A
          digraph G&nbsp;=&nbsp;(V,&nbsp;E) is equivalent to a relation E
          on V (we neglect that the version of directed graphs
          provided by the <c>digraph</c> module allows multiple edges
          between vertices). If the digraph has no cycles of length two or
          more, the reflexive and transitive closure of E is a
          partial ordering.</p>
      </item>
      <item>
        <p>A <marker id="subgraph"></marker><em>subgraph</em> G' of G is a
          digraph whose vertices and edges form subsets of the vertices
          and edges of G.</p>
      </item>
      <item>
        <p>G' is <em>maximal</em> with respect to a property P if all other
          subgraphs that include the vertices of G' do not have property P.</p>
      </item>
      <item>
        <p>A <marker id="strong_components"></marker><em>strongly connected
          component</em> is a maximal subgraph such that there is a path
          between each pair of vertices.</p>
      </item>
      <item>
        <p>A <marker id="components"></marker><em>connected component</em>
          is a maximal subgraph such that there is a path between each pair of
          vertices, considering all edges undirected.</p>
      </item>
      <item>
        <p>An <marker id="arborescence"></marker><em>arborescence</em> is an
          acyclic digraph with a vertex V, the
          <marker id="root"></marker><em>root</em>, such that there is a unique
          path from V to every other vertex of G.</p>
      </item>
      <item>
        <p>A <marker id="tree"></marker><em>tree</em> is an acyclic non-empty
          digraph such that there is a unique path between every pair of
          vertices, considering all edges undirected.</p>
      </item>
    </list>
  </description>

  <funcs>
    <func>
      <name name="arborescence_root" arity="1" since=""/>
      <fsummary>Check if a digraph is an arborescence.</fsummary>
      <desc>
        <p>Returns <c>{yes, <anno>Root</anno>}</c> if <c><anno>Root</anno></c>
          is the <seeerl marker="#root">root</seeerl> of the arborescence
          <c><anno>Digraph</anno></c>, otherwise <c>no</c>.</p>
      </desc>
    </func>

    <func>
      <name name="components" arity="1" since=""/>
      <fsummary>Return the components of a digraph.</fsummary>
      <desc>
        <p>Returns a list
          of <seeerl marker="#components">connected components</seeerl>.
          Each component is represented by its
          vertices. The order of the vertices and the order of the
          components are arbitrary. Each vertex of digraph
          <c><anno>Digraph</anno></c> occurs in exactly one component.</p>
      </desc>
    </func>

    <func>
      <name name="condensation" arity="1" since=""/>
      <fsummary>Return a condensed graph of a digraph.</fsummary>
      <desc>
        <p>Creates a digraph where the vertices are
          the <seeerl marker="#strong_components">strongly connected
          components</seeerl> of <c><anno>Digraph</anno></c> as returned by
          <seemfa marker="#strong_components/1">
          <c>strong_components/1</c></seemfa>.
          If X and Y are two different strongly
          connected components, and vertices x and y exist in X
          and Y, respectively, such that there is an
          edge <seeerl marker="#emanate">emanating</seeerl> from x
          and <seeerl marker="#incident">incident</seeerl> on y, then 
          an edge emanating from X and incident on Y is created.</p>
        <p>The created digraph has the same type as <c><anno>Digraph</anno></c>.
          All vertices and edges have the
          default <seeerl marker="#label">label</seeerl> <c>[]</c>.</p>
        <p>Each <seeerl marker="#cycle">cycle</seeerl> is
          included in some strongly connected component, which implies that
          a <seeerl marker="#topsort">topological ordering</seeerl> of the
          created digraph always exists.</p>
      </desc>
    </func>

    <func>
      <name name="cyclic_strong_components" arity="1" since=""/>
      <fsummary>Return the cyclic strong components of a digraph.</fsummary>
      <desc>
        <p>Returns a list of <seeerl marker="#strong_components">strongly
          connected components</seeerl>. 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 <seeerl marker="#cycle">cycle</seeerl> in
          <c><anno>Digraph</anno></c> are returned, otherwise the returned
          list is equal to that returned by
          <seemfa marker="#strong_components/1">
          <c>strong_components/1</c></seemfa>.</p>
      </desc>
    </func>

    <func>
      <name name="is_acyclic" arity="1" since=""/>
      <fsummary>Check if a digraph is acyclic.</fsummary>
      <desc>
        <p>Returns <c>true</c> if and only if digraph
          <c><anno>Digraph</anno></c> is
          <seeerl marker="#acyclic_digraph">acyclic</seeerl>.</p>
      </desc>
    </func>

    <func>
      <name name="is_arborescence" arity="1" since=""/>
      <fsummary>Check if a digraph is an arborescence.</fsummary>
      <desc>
        <p>Returns <c>true</c> if and only if digraph
          <c><anno>Digraph</anno></c> is
	  an <seeerl marker="#arborescence">arborescence</seeerl>.</p>
      </desc>
    </func>

    <func>
      <name name="is_tree" arity="1" since=""/>
      <fsummary>Check if a digraph is a tree.</fsummary>
      <desc>
        <p>Returns <c>true</c> if and only if digraph
          <c><anno>Digraph</anno></c> is
          a <seeerl marker="#tree">tree</seeerl>.</p>
      </desc>
    </func>

    <func>
      <name name="loop_vertices" arity="1" since=""/>
      <fsummary>Return the vertices of a digraph included in some loop.
      </fsummary>
      <desc>
        <p>Returns a list of all vertices of <c><anno>Digraph</anno></c> that
          are included in some <seeerl marker="#loop">loop</seeerl>.</p>
      </desc>
    </func>

    <func>
      <name name="postorder" arity="1" since=""/>
      <fsummary>Return the vertices of a digraph in postorder.</fsummary>
      <desc>
        <p>Returns all vertices of digraph <c><anno>Digraph</anno></c>.
          The order is given by
          a <seeerl marker="#depth_first_traversal">depth-first
          traversal</seeerl> 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.</p>
      </desc>
    </func>

    <func>
      <name name="preorder" arity="1" since=""/>
      <fsummary>Return the vertices of a digraph in preorder.</fsummary>
      <desc>
        <p>Returns all vertices of digraph <c><anno>Digraph</anno></c>.
          The order is given by
          a <seeerl marker="#depth_first_traversal">depth-first
          traversal</seeerl> of the digraph, collecting visited
          vertices in preorder.</p>
      </desc>
    </func>

    <func>
      <name name="reachable" arity="2" since=""/>
      <fsummary>Return the vertices reachable from some vertices of a digraph.
      </fsummary>
      <desc>
        <p>Returns an unsorted list of digraph vertices such that for
          each vertex in the list, there is a
          <seeerl marker="#path">path</seeerl> in <c><anno>Digraph</anno></c>
          from some
          vertex of <c><anno>Vertices</anno></c> to the vertex. In particular,
          as paths can have length zero, the vertices of
          <c><anno>Vertices</anno></c> are included in the returned list.</p>
      </desc>
    </func>

    <func>
      <name name="reachable_neighbours" arity="2" since=""/>
      <fsummary>Return the neighbors reachable from some vertices of a
        digraph.</fsummary>
      <desc>
        <p>Returns an unsorted list of digraph vertices such that for
          each vertex in the list, there is a
          <seeerl marker="#path">path</seeerl> in <c><anno>Digraph</anno></c>
          of length
          one or more from some vertex of <c><anno>Vertices</anno></c> to the
          vertex. As a consequence, only those vertices
          of <c><anno>Vertices</anno></c> that are included in
          some <seeerl marker="#cycle">cycle</seeerl> are returned.</p>
      </desc>
    </func>

    <func>
      <name name="reaching" arity="2" since=""/>
      <fsummary>Return the vertices that reach some vertices of a digraph.
      </fsummary>
      <desc>
        <p>Returns an unsorted list of digraph vertices such that for
          each vertex in the list, there is
          a <seeerl marker="#path">path</seeerl> from the vertex to some
          vertex of <c><anno>Vertices</anno></c>. In particular, as paths
          can have length zero, the vertices of <c><anno>Vertices</anno></c>
          are included in the returned list.</p>
      </desc>
    </func>

    <func>
      <name name="reaching_neighbours" arity="2" since=""/>
      <fsummary>Return the neighbors that reach some vertices of a digraph.
      </fsummary>
      <desc>
        <p>Returns an unsorted list of digraph vertices such that for
          each vertex in the list, there is
          a <seeerl marker="#path">path</seeerl> of length one or more
          from the vertex to some vertex of <c><anno>Vertices</anno></c>.
          Therefore only those vertices of <c><anno>Vertices</anno></c>
          that are included
          in some <seeerl marker="#cycle">cycle</seeerl> are returned.</p>
      </desc>
    </func>

    <func>
      <name name="strong_components" arity="1" since=""/>
      <fsummary>Return the strong components of a digraph.</fsummary>
      <desc>
        <p>Returns a list of <seeerl marker="#strong_components">strongly
          connected components</seeerl>.
          Each strongly component is represented
          by its vertices. The order of the vertices and the order of
          the components are arbitrary. Each vertex of digraph
          <c><anno>Digraph</anno></c> occurs in exactly one strong component.
        </p>
      </desc>
    </func>

    <func>
      <name name="subgraph" arity="2" since=""/>
      <name name="subgraph" arity="3" since=""/>
      <fsummary>Return a subgraph of a digraph.</fsummary>
      <desc>
        <p>Creates a maximal <seeerl marker="#subgraph">subgraph</seeerl>
          of <c>Digraph</c> having
          as vertices those vertices of <c><anno>Digraph</anno></c> that are
          mentioned in <c><anno>Vertices</anno></c>.</p>
        <p>If the value of option <c>type</c> is <c>inherit</c>, which is
          the default, the type of <c><anno>Digraph</anno></c> is used
          for the subgraph as well. Otherwise the option value of <c>type</c>
          is used as argument to
        <seemfa marker="digraph#new/1"><c>digraph:new/1</c></seemfa>.</p>
        <p>If the value of option <c>keep_labels</c> is <c>true</c>,
          which is the default,
          the <seeerl marker="#label">labels</seeerl> of vertices and edges
          of <c><anno>Digraph</anno></c> are used for the subgraph as well. If
          the value is <c>false</c>, default label <c>[]</c> is used
          for the vertices and edges of the subgroup.</p>
        <p><c>subgraph(<anno>Digraph</anno>, <anno>Vertices</anno>)</c> is
          equivalent to
          <c>subgraph(<anno>Digraph</anno>, <anno>Vertices</anno>, [])</c>.</p>
        <p>If any of the arguments are invalid, a <c>badarg</c> exception is
          raised.</p>
      </desc>
    </func>

    <func>
      <name name="topsort" arity="1" since=""/>
      <fsummary>Return a topological sorting of the vertices of a digraph.
      </fsummary>
      <desc>
        <p>Returns a <seeerl marker="#topsort">topological
          ordering</seeerl> of the vertices of digraph
          <c><anno>Digraph</anno></c> if such an ordering exists, otherwise
          <c>false</c>. For each vertex in the returned list,
          no <seeerl marker="#out_neighbour">out-neighbors</seeerl>
          occur earlier in the list.</p>
      </desc>
    </func>
  </funcs>

  <section>
    <title>See Also</title>
    <p><seeerl marker="digraph"><c>digraph(3)</c></seeerl></p>
  </section>
</erlref>

