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diff --git a/lib/stdlib/doc/src/digraph_utils.xml b/lib/stdlib/doc/src/digraph_utils.xml
index e481711c50..cb316e5b93 100644
--- a/lib/stdlib/doc/src/digraph_utils.xml
+++ b/lib/stdlib/doc/src/digraph_utils.xml
@@ -24,100 +24,132 @@
<title>digraph_utils</title>
<prepared>Hans Bolinder</prepared>
- <responsible>nobody</responsible>
+ <responsible></responsible>
<docno></docno>
- <approved>nobody</approved>
- <checked>no</checked>
+ <approved></approved>
+ <checked></checked>
<date>2001-08-27</date>
<rev>PA1</rev>
- <file>digraph_utils.sgml</file>
+ <file>digraph_utils.xml</file>
</header>
<module>digraph_utils</module>
- <modulesummary>Algorithms for Directed Graphs</modulesummary>
+ <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,&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>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&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.
- 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 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>This module provides algorithms based on depth-first traversal of
+ directed graphs. For basic functions on directed graphs, see the
+ <seealso marker="digraph"><c>digraph(3)</c></seealso> module.</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&nbsp;=&nbsp;(V,&nbsp;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>
+ <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>
@@ -125,237 +157,253 @@
<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>
+ <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>, otherwise <c>no</c>.</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
+ <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>
+ 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"/>
<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 two different 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>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
+ <seealso marker="#strong_components/1">
+ <c>strong_components/1</c></seealso>.
+ 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 <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>
+ All vertices and edges have the
+ default <seealso marker="#label">label</seealso> <c>[]</c>.</p>
+ <p>Each <seealso marker="#cycle">cycle</seealso> is
+ included in some strongly connected component, which implies that
+ a <seealso marker="#topsort">topological ordering</seealso> of the
+ created digraph always exists.</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
+ <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>
+ <c><anno>Digraph</anno></c> are returned, otherwise the returned
+ list is equal to that returned by
+ <seealso marker="#strong_components/1">
+ <c>strong_components/1</c></seealso>.</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>
+ <p>Returns <c>true</c> if and only if 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
+ <p>Returns <c>true</c> if and only if 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
+ <p>Returns <c>true</c> if and only if digraph
<c><anno>Digraph</anno></c> is
- a <seealso marker="#tree">tree</seealso>.</p>
+ 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>
+ <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>
+ <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>
+ <fsummary>Return the vertices of a digraph in postorder.</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
+ <p>Returns all vertices of 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>
+ 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>
+ <fsummary>Return the vertices of a digraph in preorder.</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>
+ <p>Returns all vertices of 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 preorder.</p>
</desc>
</func>
+
<func>
<name name="reachable" arity="2"/>
- <fsummary>Return the vertices reachable from some vertices of a digraph.</fsummary>
+ <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
+ 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>
+ 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"/>
- <fsummary>Return the neighbours reachable from some vertices of a digraph.</fsummary>
+ <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 <seealso marker="#path">path</seealso> in <c><anno>Digraph</anno></c> of length
+ 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>
+ 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>
+ <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>
+ 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, 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"/>
- <fsummary>Return the neighbours that reach some vertices of a digraph.</fsummary>
+ <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 <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>
+ 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>.
+ Therefore 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
+ <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
+ the components are arbitrary. Each vertex of digraph
<c><anno>Digraph</anno></c> occurs in exactly one strong component.
- </p>
+ </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
+ <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
+ 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 <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>
+ is used as argument to
+ <seealso marker="digraph:new/1"><c>digraph:new/1</c></seealso>.</p>
+ <p>If the value of option <c>keep_labels</c> is <c>true</c>,
+ which is the default,
+ 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>, 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"/>
- <fsummary>Return a topological sorting of the vertices of a digraph.</fsummary>
+ <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>
+ <p>Returns a <seealso marker="#topsort">topological
+ ordering</seealso> 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 <seealso marker="#out_neighbour">out-neighbors</seealso>
+ occur earlier in the list.</p>
</desc>
</func>
</funcs>
<section>
<title>See Also</title>
- <p><seealso marker="digraph">digraph(3)</seealso></p>
+ <p><seealso marker="digraph"><c>digraph(3)</c></seealso></p>
</section>
</erlref>