This is the manual for version &VERSION; of the &Digraphs; package.
This package was developed at the University of St Andrews by:
<List>
<Item>Jan De Beule,</Item>
<Item>Julius Jonušas,</Item>
<Item>James D. Mitchell,</Item>
<Item>Maria Tsalakou,</Item>
<Item>Wilf A. Wilson, and</Item>
<Item>Michael C. Young.</Item>
</List>
Additional contributions were made by many people, for which the authors are
very grateful. A full list can be found on the title page.
The &Digraphs; package contains a variety of methods for efficiently creating
and storing mutable and immutable digraphs and computing information about
them. Full explanations of all the functions contained in the package are
provided below. <P/>
If the &GRAPE; package is available, it will be loaded automatically.
Digraphs created with the &Digraphs; package can be converted to &GRAPE;
graphs with <Ref Oper="Graph"/>, and conversely &GRAPE; graphs can be
converted to &Digraphs; objects with <Ref Oper="Digraph"/>. &GRAPE; is not
required for &Digraphs; to run. <P/>
The &BLISS; tool <Cite Key="JK07"/> is included in this package. It
is an open-source tool for computing automorphism groups and canonical forms
of graphs, written by Tommi Junttila and Petteri Kaski. Several of the methods
in the &Digraphs; package rely on &BLISS;. If the &NautyTracesInterface;
package for GAP is available then it is also possible to use &NAUTY; <Cite
Key="MP14"/> for computing automorphism groups and canonical forms
in &Digraphs;. See Section
<Ref Sect="Isomorphisms and canonical labellings"/> for more details.
<P/>
The &EDGE_PLANARITY_SUITE; is also included in &Digraphs;; see
<Cite Key="BM04"/>, <Cite Key="B06"/>, <Cite Key="BM06"/>, and <Cite
Key="B12"/> . The &EDGE_PLANARITY_SUITE; is an open-source
implementation of the edge addition planar graph embedding algorithm and
related algorithms by John M. Boyer. See Section <Ref Sect="Planarity"/> for
more details.
<P/>
From version 1.0.0 of this package, digraphs can be either mutable or immutable.
Mutable digraphs can be changed in-place by many of the methods in the
package, which avoids unnecessary copying. Immutable digraphs cannot be
changed in-place, but their advantage is that the value of an attribute of
an immutable digraph is only ever computed once. Mutable digraphs can be
converted into immutable digraphs in-place using <Ref Func="MakeImmutable"
BookName = "ref"/>. One of the motivations for introducing mutable
digraphs in version 1.0.0 was that in practice the authors often wanted
to create a digraph and immediately modify it (removing certain edges,
loops, and so on). Before version 1.0.0, this involved copying the digraph
several times, with each copy being discarded almost immediately.
From version 1.0.0, this unnecessary copying can be eliminated by
first creating a mutable digraph, then changing it in-place, and finally
converting the mutable digraph to an immutable one in-place
(if desirable).
For the purposes of this package and its documentation, the following
definitions apply: <P/>
A <E>digraph</E> <M>E=(E^0,E^1,r,s)</M>, also known as a <E>directed
graph</E>, consists of a set of vertices <M>E^0</M> and a set of edges
<M>E^1</M> together with functions <M>s, r: E^1 \to E^0</M>, called the
<E>source</E> and <E>range</E>, respectively.
The source and range of an edge is respectively the values of <M>s, r</M> at
that edge.
An edge is called a <E>loop</E> if its source and range are the same.
A digraph is called a <E>multidigraph</E> if there exist two or more edges
with the same source and the same range. <P/>
A <E>directed walk</E> on a digraph is a sequence of alternating vertices
and edges <M>(v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)</M> such that each edge
<M>e_i</M> has source <M>v_i</M> and range <M>v_{i+1}</M>. A <E>directed
path</E> is a directed walk where no vertex (and hence no edge) is
repeated. A <E>directed circuit</E> is a directed walk where <M>v_1 =
v_n</M>, and a <E>directed cycle</E> is a directed circuit where where no
vertex is repeated, except for <M>v_1 = v_n</M>. <P/>
The <E>length</E> of a directed walk <M>(v_1, e_1, v_2, e_2, ..., e_{n-1},
v_n)</M> is equal to <M>n-1</M>, the number of edges it contains. A
directed walk (or path) <M>(v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)</M> is
sometimes called a directed walk (or path) <E>from vertex <M>v_1</M> to
vertex <M>v_n</M></E>. A directed walk of zero length, i.e. a sequence
<M>(v)</M> for some vertex <M>v</M>, is called <E>trivial</E>. A trivial
directed walk is considered to be both a circuit and a cycle, as is the
empty directed walk <M>()</M>. A <E>simple circuit</E> is another name for a
non-trivial and non-empty directed cycle.<P/>
</Subsection>
</Section>
</Chapter>
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