<?xml version="1.0" encoding="latin1" ?> <!DOCTYPE erlref SYSTEM "erlref.dtd"> <erlref> <header> <copyright> <year>2000</year><year>2011</year> <holder>Ericsson AB. All Rights Reserved.</holder> </copyright> <legalnotice> The contents of this file are subject to the Erlang Public License, Version 1.1, (the "License"); you may not use this file except in compliance with the License. You should have received a copy of the Erlang Public License along with this software. If not, it can be retrieved online at http://www.erlang.org/. Software distributed under the License is distributed on an "AS IS" basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License for the specific language governing rights and limitations under the License. </legalnotice> <title>digraph_utils</title> <prepared>Hans Bolinder</prepared> <responsible>nobody</responsible> <docno></docno> <approved>nobody</approved> <checked>no</checked> <date>2001-08-27</date> <rev>PA1</rev> <file>digraph_utils.sgml</file> </header> <module>digraph_utils</module> <modulesummary>Algorithms for Directed Graphs</modulesummary> <description> <p>The <c>digraph_utils</c> module implements some algorithms based on depth-first traversal of directed graphs. See the <c>digraph</c> module for basic functions on directed graphs. </p> <p>A <marker id="digraph"></marker><em>directed graph</em> (or just "digraph") is a pair (V, 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 × V (the Cartesian product of V with itself). </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>.</p> <p>An edge e = (v, 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. 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, 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 <marker id="length"></marker><em>length</em> of the path P is k-1. 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. An <marker id="acyclic_digraph"></marker><em>acyclic digraph</em> is a digraph that has no cycles. </p> <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 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. </p> <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. 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 = (V, E) is equivalent to a relation E on V (we neglect the fact that the version of directed graphs implemented in the <c>digraph</c> 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. </p> <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. G' is <em>maximal</em> with respect to a property P if all other subgraphs that include the vertices of G' do not have the property 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. 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. 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. 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> </description> <datatypes> <datatype> <name><marker id="type-digraph">digraph()</marker></name> <desc><p>A digraph as returned by <c>digraph:new/0,1</c>.</p></desc> </datatype> </datatypes> <funcs> <func> <name name="arborescence_root" arity="1"/> <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 <seealso marker="#root">root</seealso> of the arborescence <c><anno>Digraph</anno></c>, <c>no</c> otherwise. </p> </desc> </func> <func> <name name="components" arity="1"/> <fsummary>Return the components of a digraph.</fsummary> <desc> <p>Returns a list of <seealso marker="#components">connected components</seealso>. 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 <c><anno>Digraph</anno></c> occurs in exactly one component. </p> </desc> </func> <func> <name name="condensation" arity="1"/> <fsummary>Return a condensed graph of a digraph.</fsummary> <desc> <p>Creates a digraph where the vertices are the <seealso marker="#strong_components">strongly connected components</seealso> of <c><anno>Digraph</anno></c> as returned by <c>strong_components/1</c>. 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 <seealso marker="#emanate">emanating</seealso> from x and <seealso marker="#incident">incident</seealso> 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 <seealso marker="#label">label</seealso> <c>[]</c>. </p> <p>Each and every <seealso marker="#cycle">cycle</seealso> is included in some strongly connected component, which implies that there always exists a <seealso marker="#topsort">topological ordering</seealso> of the created digraph.</p> </desc> </func> <func> <name name="cyclic_strong_components" arity="1"/> <fsummary>Return the cyclic strong components of a digraph.</fsummary> <desc> <p>Returns a list of <seealso marker="#strong_components">strongly connected components</seealso>. 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 <seealso marker="#cycle">cycle</seealso> in <c><anno>Digraph</anno></c> are returned, otherwise the returned list is equal to that returned by <c>strong_components/1</c>. </p> </desc> </func> <func> <name name="is_acyclic" arity="1"/> <fsummary>Check if a digraph is acyclic.</fsummary> <desc> <p>Returns <c>true</c> if and only if the digraph <c><anno>Digraph</anno></c> is <seealso marker="#acyclic_digraph">acyclic</seealso>.</p> </desc> </func> <func> <name name="is_arborescence" arity="1"/> <fsummary>Check if a digraph is an arborescence.</fsummary> <desc> <p>Returns <c>true</c> if and only if the digraph <c><anno>Digraph</anno></c> is an <seealso marker="#arborescence">arborescence</seealso>.</p> </desc> </func> <func> <name name="is_tree" arity="1"/> <fsummary>Check if a digraph is a tree.</fsummary> <desc> <p>Returns <c>true</c> if and only if the digraph <c><anno>Digraph</anno></c> is a <seealso marker="#tree">tree</seealso>.</p> </desc> </func> <func> <name name="loop_vertices" arity="1"/> <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 <seealso marker="#loop">loop</seealso>.</p> </desc> </func> <func> <name name="postorder" arity="1"/> <fsummary>Return the vertices of a digraph in post-order.</fsummary> <desc> <p>Returns all vertices of the digraph <c><anno>Digraph</anno></c>. The order is given by a <seealso marker="#depth_first_traversal">depth-first traversal</seealso> 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"/> <fsummary>Return the vertices of a digraph in pre-order.</fsummary> <desc> <p>Returns all vertices of the digraph <c><anno>Digraph</anno></c>. The order is given by a <seealso marker="#depth_first_traversal">depth-first traversal</seealso> of the digraph, collecting visited vertices in pre-order.</p> </desc> </func> <func> <name name="reachable" arity="2"/> <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 <seealso marker="#path">path</seealso> in <c><anno>Digraph</anno></c> from some vertex of <c><anno>Vertices</anno></c> to the vertex. In particular, since paths may 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"/> <fsummary>Return the neighbours 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 <seealso marker="#path">path</seealso> 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 <seealso marker="#cycle">cycle</seealso> are returned. </p> </desc> </func> <func> <name name="reaching" arity="2"/> <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 <seealso marker="#path">path</seealso> from the vertex to some vertex of <c><anno>Vertices</anno></c>. In particular, since paths may 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"/> <fsummary>Return the neighbours 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 <seealso marker="#path">path</seealso> of length one or more from the vertex to some vertex of <c><anno>Vertices</anno></c>. As a consequence, only those vertices of <c><anno>Vertices</anno></c> that are included in some <seealso marker="#cycle">cycle</seealso> are returned. </p> </desc> </func> <func> <name name="strong_components" arity="1"/> <fsummary>Return the strong components of a digraph.</fsummary> <desc> <p>Returns a list of <seealso marker="#strong_components">strongly connected components</seealso>. 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 <c><anno>Digraph</anno></c> occurs in exactly one strong component. </p> </desc> </func> <func> <name name="subgraph" arity="2"/> <name name="subgraph" arity="3"/> <fsummary>Return a subgraph of a digraph.</fsummary> <desc> <p>Creates a maximal <seealso marker="#subgraph">subgraph</seealso> 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 the option <c>type</c> is <c>inherit</c>, which is the default, then 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 <c>digraph:new/1</c>. </p> <p>If the value of the option <c>keep_labels</c> is <c>true</c>, which is the default, then the <seealso marker="#label">labels</seealso> of vertices and edges of <c><anno>Digraph</anno></c> are used for the subgraph as well. If the value is <c>false</c>, then the default label, <c>[]</c>, is used for the subgraph's vertices and edges. </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>There will be a <c>badarg</c> exception if any of the arguments are invalid. </p> </desc> </func> <func> <name name="topsort" arity="1"/> <fsummary>Return a topological sorting of the vertices of a digraph.</fsummary> <desc> <p>Returns a <seealso marker="#topsort">topological ordering</seealso> of the vertices of the digraph <c><anno>Digraph</anno></c> if such an ordering exists, <c>false</c> otherwise. For each vertex in the returned list, there are no <seealso marker="#out_neighbour">out-neighbours</seealso> that occur earlier in the list.</p> </desc> </func> </funcs> <section> <title>See Also</title> <p><seealso marker="digraph">digraph(3)</seealso></p> </section> </erlref>