<html><head><title>[grape] 6 Functions to construct new graphs from old</title></head>
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<h1>6 Functions to construct new graphs from old</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP006.htm#SECT001">InducedSubgraph</a>
<li> <A HREF="CHAP006.htm#SECT002">DistanceSetInduced</a>
<li> <A HREF="CHAP006.htm#SECT003">DistanceGraph</a>
<li> <A HREF="CHAP006.htm#SECT004">ComplementGraph</a>
<li> <A HREF="CHAP006.htm#SECT005">PointGraph</a>
<li> <A HREF="CHAP006.htm#SECT006">EdgeGraph</a>
<li> <A HREF="CHAP006.htm#SECT007">SwitchedGraph</a>
<li> <A HREF="CHAP006.htm#SECT008">UnderlyingGraph</a>
<li> <A HREF="CHAP006.htm#SECT009">QuotientGraph</a>
<li> <A HREF="CHAP006.htm#SECT010">BipartiteDouble</a>
<li> <A HREF="CHAP006.htm#SECT011">GeodesicsGraph</a>
<li> <A HREF="CHAP006.htm#SECT012">CollapsedIndependentOrbitsGraph</a>
<li> <A HREF="CHAP006.htm#SECT013">CollapsedCompleteOrbitsGraph</a>
<li> <A HREF="CHAP006.htm#SECT014">NewGroupGraph</a>
</ol><p>
<p>
This chapter describes functions to construct new graphs from old
ones.
<p>
<p>
<h2><a name="SECT001">6.1 InducedSubgraph</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>InducedSubgraph( </code><var>gamma</var><code>, </code><var>V</var><code> )</code>
<li><code>InducedSubgraph( </code><var>gamma</var><code>, </code><var>V</var><code>, </code><var>G</var><code> )</code>
<p>
This function returns the subgraph of <var>gamma</var> induced on the vertex
list <var>V</var> (which must not contain repeated elements). If the optional
third parameter <var>G</var> is given, then it is assumed that <var>G</var> fixes <var>V</var>
setwise, and is a group of automorphisms of the induced subgraph when
restricted to <var>V</var>. In that case, the image of <var>G</var> acting on <var>V</var> is the
group associated with the induced subgraph. If no such <var>G</var> is given then
the associated group is trivial. The name of vertex <var>i</var> in the induced
subgraph is equal to the name of vertex <code></code><var>V</var><code>[</code><var>i</var><code>]</code> in <var>gamma</var>.
<p>
<pre>
gap> gamma := JohnsonGraph(4,2);;
gap> S := [2,3,4,5];;
gap> square := InducedSubgraph( gamma, S, Stabilizer(gamma.group,S,OnSets) );
rec(
isGraph := true,
order := 4,
group := Group( [ (1,4), (1,3)(2,4), (1,2)(3,4) ] ),
schreierVector := [ -1, 3, 2, 1 ],
adjacencies := [ [ 2, 3 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ] ] )
gap> GlobalParameters(square);
[ [ 0, 0, 2 ], [ 1, 0, 1 ], [ 2, 0, 0 ] ]
</pre>
<p>
<p>
<h2><a name="SECT002">6.2 DistanceSetInduced</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>DistanceSetInduced( </code><var>gamma</var><code>, </code><var>distances</var><code>, </code><var>V</var><code> )</code>
<li><code>DistanceSetInduced( </code><var>gamma</var><code>, </code><var>distances</var><code>, </code><var>V</var><code>, </code><var>G</var><code> )</code>
<p>
Let <var>V</var> be a vertex or a nonempty list of vertices of <var>gamma</var>.
This function returns the subgraph of <var>gamma</var> induced on the set of
vertices <var>w</var> of <var>gamma</var> such that <var>d(<var>V</var>,w)</var> is in <var>distances</var> (a list
or singleton distance).
<p>
The optional parameter <var>G</var>, if present, is assumed to be a subgroup of
<var>Aut(<var>gamma</var>)</var> fixing <var>V</var> setwise. Including such a <var>G</var> can speed up
the function.
<p>
See also <a href="CHAP003.htm#SSEC015.1">Distance</a> and <a href="CHAP005.htm#SSEC004.1">DistanceSet</a>.
<p>
<pre>
gap> DistanceSetInduced( JohnsonGraph(4,2), [0,1], [1] );
rec(
isGraph := true,
order := 5,
group := Group( [ (2,3)(4,5), (2,5)(3,4) ] ),
schreierVector := [ -1, -2, 1, 2, 2 ],
adjacencies := [ [ 2, 3, 4, 5 ], [ 1, 3, 4 ] ],
representatives := [ 1, 2 ],
isSimple := true,
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ] ] )
</pre>
<p>
<p>
<h2><a name="SECT003">6.3 DistanceGraph</a></h2>
<p><p>
<a name = "SSEC003.1"></a>
<li><code>DistanceGraph( </code><var>gamma</var><code>, </code><var>distances</var><code> )</code>
<p>
This function returns the graph <var>delta</var>, with the same vertex-set
(and vertex-names) as <var>gamma</var>, such that <var>[x,y]</var> is an edge of <var>delta</var>
if and only if <var>d(x,y)</var> (in <var>gamma</var>) is in <var>distances</var> (a list or
singleton distance).
<p>
<pre>
gap> DistanceGraph( JohnsonGraph(4,2), [2] );
rec(
isGraph := true,
order := 6,
group := Group( [ (1,4,6,3)(2,5), (2,4)(3,5) ] ),
schreierVector := [ -1, 2, 1, 1, 1, 1 ],
adjacencies := [ [ 6 ] ],
representatives := [ 1 ],
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
isSimple := true )
gap> ConnectedComponents(last);
[ [ 1, 6 ], [ 2, 5 ], [ 3, 4 ] ]
</pre>
<p>
<p>
<h2><a name="SECT004">6.4 ComplementGraph</a></h2>
<p><p>
<a name = "SSEC004.1"></a>
<li><code>ComplementGraph( </code><var>gamma</var><code> )</code>
<li><code>ComplementGraph( </code><var>gamma</var><code>, </code><var>comploops</var><code> )</code>
<p>
This function returns the complement of the graph <var>gamma</var>. The optional
boolean parameter <var>comploops</var> determines whether or not loops/nonloops are
complemented (default: <code>false</code> (loops/nonloops are not complemented)). The
returned graph will have the same vertex-names as <var>gamma</var>.
<p>
<pre>
gap> ComplementGraph( NullGraph(SymmetricGroup(3)) );
rec(
isGraph := true,
order := 3,
group := SymmetricGroup( [ 1 .. 3 ] ),
schreierVector := [ -1, 1, 1 ],
adjacencies := [ [ 2, 3 ] ],
representatives := [ 1 ],
isSimple := true )
gap> IsLoopy(last);
false
gap> IsLoopy(ComplementGraph(NullGraph(SymmetricGroup(3)),true));
true
</pre>
<p>
<p>
<h2><a name="SECT005">6.5 PointGraph</a></h2>
<p><p>
<a name = "SSEC005.1"></a>
<li><code>PointGraph( </code><var>gamma</var><code> )</code>
<li><code>PointGraph( </code><var>gamma</var><code>, </code><var>v</var><code> )</code>
<p>
Assuming that <var>gamma</var> is simple, connected, and bipartite, this function
returns the induced subgraph on the connected component of
<code>DistanceGraph(</code><var>gamma</var><code>,2)</code> containing the vertex <var>v</var> (default:
<var><var>v</var>=1</var>).
<p>
Thus, if <var>gamma</var> is the incidence graph of a connected geometry of rank
2, and <var>v</var> represents a point, then the point graph of the geometry is
returned.
<p>
<pre>
gap> BipartiteDouble( CompleteGraph(SymmetricGroup(4)) );;
gap> PointGraph(last);
rec(
isGraph := true,
order := 4,
group := Group( [ (1,2), (1,2,3,4) ] ),
schreierVector := [ -1, 1, 2, 2 ],
adjacencies := [ [ 2, 3, 4 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ 1, "+" ], [ 2, "+" ], [ 3, "+" ], [ 4, "+" ] ] )
gap> IsCompleteGraph(last);
true
</pre>
<p>
<p>
<h2><a name="SECT006">6.6 EdgeGraph</a></h2>
<p><p>
<a name = "SSEC006.1"></a>
<li><code>EdgeGraph( </code><var>gamma</var><code> )</code>
<p>
This function return a graph isomorphic to the the edge graph (also
called the line graph) of the simple graph <var>gamma</var>.
<p>
This <strong>edge graph</strong> <var>delta</var> has the unordered edges of <var>gamma</var>
as vertices, and <var>e</var> is joined to <var>f</var> in <var>delta</var> precisely when
<var>|ecapf|=1</var>. The name of the vertex of the returned graph
corresponding to the unordered edge <var>[v,w]</var> of <var>gamma</var> (with <var>v< w</var>)
is <code>[VertexName(</code><var>gamma</var><code>,</code><var>v</var><code>),VertexName(</code><var>gamma</var><code>,</code><var>w</var><code>)]</code>.
<p>
<pre>
gap> EdgeGraph( CompleteGraph(SymmetricGroup(5)) );
rec(
isGraph := true,
order := 10,
group := Group( [ ( 1, 5, 8,10, 4)( 2, 6, 9, 3, 7), ( 2, 5)( 3, 6)( 4, 7)
] ),
schreierVector := [ -1, 2, 2, 1, 1, 1, 2, 1, 1, 1 ],
adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
[ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ] )
gap> GlobalParameters(last);
[ [ 0, 0, 6 ], [ 1, 3, 2 ], [ 4, 2, 0 ] ]
</pre>
<p>
<p>
<h2><a name="SECT007">6.7 SwitchedGraph</a></h2>
<p><p>
<a name = "SSEC007.1"></a>
<li><code>SwitchedGraph( </code><var>gamma</var><code>, </code><var>V</var><code> )</code>
<li><code>SwitchedGraph( </code><var>gamma</var><code>, </code><var>V</var><code>, </code><var>H</var><code> )</code>
<p>
This function returns the switched graph <var>delta</var> of the graph <var>gamma</var>,
switched with respect to the vertex list (or singleton vertex) <var>V</var>.
<p>
The returned graph <var>delta</var> has vertex-set (and vertex-names) the same
as <var>gamma</var>. If vertices <var>x,y</var> of <var>delta</var> are both in <var>V</var> or both not
in <var>V</var>, then <var>[x,y]</var> is an edge of <var>delta</var> if and only if <var>[x,y]</var> is an
edge of <var>gamma</var>; otherwise <var>[x,y]</var> is an edge of <var>delta</var> if and only if
<var>[x,y]</var> is not an edge of <var>gamma</var>. If the optional third argument <var>H</var>
is given, then it is assumed to be a subgroup of Aut(<var>gamma</var>) stabilizing
<var>V</var> setwise.
<p>
<pre>
gap> J:=JohnsonGraph(4,2);
rec(
isGraph := true,
order := 6,
group := Group( [ (1,4,6,3)(2,5), (2,4)(3,5) ] ),
schreierVector := [ -1, 2, 1, 1, 1, 1 ],
adjacencies := [ [ 2, 3, 4, 5 ] ],
representatives := [ 1 ],
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
isSimple := true )
gap> S:=SwitchedGraph(J,[1,6]);
rec(
isGraph := true,
order := 6,
group := Group( () ),
schreierVector := [ -1, -2, -3, -4, -5, -6 ],
adjacencies := [ [ ], [ 3, 4 ], [ 2, 5 ], [ 2, 5 ], [ 3, 4 ], [ ] ],
representatives := [ 1, 2, 3, 4, 5, 6 ],
isSimple := true,
names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ] )
gap> ConnectedComponents(S);
[ [ 1 ], [ 2, 3, 4, 5 ], [ 6 ] ]
</pre>
<p>
<h2><a name="SECT008">6.8 UnderlyingGraph</a></h2>
<p><p>
<a name = "SSEC008.1"></a>
<li><code>UnderlyingGraph( </code><var>gamma</var><code> )</code>
<p>
This function returns the underlying graph <var>delta</var> of <var>gamma</var>. The graph
<var>delta</var> has the same vertex-set (and vertex-names) as <var>gamma</var>, and has
an edge <var>[x,y]</var> precisely when <var>gamma</var> has an edge <var>[x,y]</var> or an edge
<var>[y,x]</var>. This function also sets the <code>isSimple</code> components of <var>gamma</var>
and <var>delta</var>.
<p>
<pre>
gap> gamma := EdgeOrbitsGraph( Group((1,2,3,4)), [1,2] );
rec(
isGraph := true,
order := 4,
group := Group( [ (1,2,3,4) ] ),
schreierVector := [ -1, 1, 1, 1 ],
adjacencies := [ [ 2 ] ],
representatives := [ 1 ],
isSimple := false )
gap> UnderlyingGraph(gamma);
rec(
isGraph := true,
order := 4,
group := Group( [ (1,2,3,4) ] ),
schreierVector := [ -1, 1, 1, 1 ],
adjacencies := [ [ 2, 4 ] ],
representatives := [ 1 ],
isSimple := true )
</pre>
<p>
<p>
<h2><a name="SECT009">6.9 QuotientGraph</a></h2>
<p><p>
<a name = "SSEC009.1"></a>
<li><code>QuotientGraph( </code><var>gamma</var><code>, </code><var>R</var><code> )</code>
<p>
Let <var>S</var> be the smallest <code></code><var>gamma</var><code>.group</code>-invariant equivalence relation
on the vertices of <var>gamma</var>, such that <var>S</var> contains the relation <var>R</var>
(which should be a list of ordered pairs (length 2 lists) of vertices
of <var>gamma</var>). Then this function returns a graph isomorphic to the
quotient <var>delta</var> of the graph <var>gamma</var>, defined as follows. The vertices
of <var>delta</var> are the equivalence classes of <var>S</var>, and <var>[X,Y]</var> is an edge of
<var>delta</var> if and only if <var>[x,y]</var> is an edge of <var>gamma</var> for some <var>xinX</var>,
<var>yinY</var>. The name of a vertex <var>v</var> in the returned graph is a list (not
necessarily ordered) of the vertex-names of <var>gamma</var> for the vertices in
the equivalence class corresponding to <var>v</var>.
<p>
<pre>
gap> gamma := JohnsonGraph(4,2);;
gap> QuotientGraph( gamma, [[1,6]] );
rec(
isGraph := true,
order := 3,
group := Group( [ (1,3), (2,3) ] ),
schreierVector := [ -1, 2, 1 ],
adjacencies := [ [ 2, 3 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 3 ], [ 2, 4 ] ],
[ [ 1, 4 ], [ 2, 3 ] ] ] )
gap> IsCompleteGraph(last);
true
</pre>
<p>
<p>
<h2><a name="SECT010">6.10 BipartiteDouble</a></h2>
<p><p>
<a name = "SSEC010.1"></a>
<li><code>BipartiteDouble( </code><var>gamma</var><code> )</code>
<p>
This function returns the bipartite double of the graph <var>gamma</var>, as
defined in <a href="biblio.htm#BCN89"><cite>BCN89</cite></a>.
<p>
<pre>
gap> gamma := JohnsonGraph(4,2);;
gap> IsBipartite(gamma);
false
gap> delta := BipartiteDouble(gamma);
rec(
isGraph := true,
order := 12,
group := Group( [ ( 1, 4, 6, 3)( 2, 5)( 7,10,12, 9)( 8,11),
( 2, 4)( 3, 5)( 8,10)( 9,11), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)
( 6,12) ] ),
schreierVector := [ -1, 2, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3 ],
adjacencies := [ [ 8, 9, 10, 11 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ [ 1, 2 ], "+" ], [ [ 1, 3 ], "+" ], [ [ 1, 4 ], "+" ],
[ [ 2, 3 ], "+" ], [ [ 2, 4 ], "+" ], [ [ 3, 4 ], "+" ],
[ [ 1, 2 ], "-" ], [ [ 1, 3 ], "-" ], [ [ 1, 4 ], "-" ],
[ [ 2, 3 ], "-" ], [ [ 2, 4 ], "-" ], [ [ 3, 4 ], "-" ] ] )
gap> IsBipartite(delta);
true
</pre>
<p>
<p>
<h2><a name="SECT011">6.11 GeodesicsGraph</a></h2>
<p><p>
<a name = "SSEC011.1"></a>
<li><code>GeodesicsGraph( </code><var>gamma</var><code>, </code><var>x</var><code>, </code><var>y</var><code> )</code>
<p>
This function returns the the graph induced on the set of geodesics
in <var>gamma</var> between the vertices <var>x</var> and <var>y</var>, but including neither <var>x</var>
nor <var>y</var>. This function is only for a simple graph <var>gamma</var>.
<p>
<pre>
gap> GeodesicsGraph( JohnsonGraph(4,2), 1, 6 );
rec(
isGraph := true,
order := 4,
group := Group( [ (1,3)(2,4), (1,4)(2,3), (2,3) ] ),
schreierVector := [ -1, 2, 1, 2 ],
adjacencies := [ [ 2, 3 ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ] ] )
gap> GlobalParameters(last);
[ [ 0, 0, 2 ], [ 1, 0, 1 ], [ 2, 0, 0 ] ]
</pre>
<p>
<p>
<h2><a name="SECT012">6.12 CollapsedIndependentOrbitsGraph</a></h2>
<p><p>
<a name = "SSEC012.1"></a>
<li><code>CollapsedIndependentOrbitsGraph( </code><var>G</var><code>, </code><var>gamma</var><code> )</code>
<li><code>CollapsedIndependentOrbitsGraph( </code><var>G</var><code>, </code><var>gamma</var><code>, </code><var>N</var><code> )</code>
<p>
Given a subgroup <var>G</var> of the automorphism group of the simple graph
<var>gamma</var>, this function returns a graph isomorphic to <var>delta</var>, defined as
follows. The vertices of <var>delta</var> are those <var>G</var>-orbits of the vertices of
<var>gamma</var> that are independent sets in <var>gamma</var>, and <var>x</var> is joined to <var>y</var> in
<var>delta</var> if and only if <var>xcupy</var> is <strong>not</strong> an independent set in <var>gamma</var>.
The name of a vertex <var>v</var> in the returned graph is a list (not necessarily
ordered) of the vertex-names of <var>gamma</var> for the vertices in the <var>G</var>-orbit
corresponding to <var>v</var>.
<p>
If the optional parameter <var>N</var> is given, then it is assumed to be a
subgroup of <var>Aut(<var>gamma</var>)</var> preserving the set of <var>G</var>-orbits of the
vertices of <var>gamma</var> (for example, the normalizer in <code></code><var>gamma</var><code>.group</code> of
<var>G</var>). This information can make the function more efficient.
<p>
<pre>
gap> G := Group( (1,2) );;
gap> gamma := NullGraph( SymmetricGroup(3) );;
gap> CollapsedIndependentOrbitsGraph( G, gamma );
rec(
isGraph := true,
order := 2,
group := Group( [ () ] ),
schreierVector := [ -1, -2 ],
adjacencies := [ [ ], [ ] ],
representatives := [ 1, 2 ],
isSimple := true,
names := [ [ 1, 2 ], [ 3 ] ] )
gap> gamma := CompleteGraph( SymmetricGroup(3) );;
gap> CollapsedIndependentOrbitsGraph( G, gamma );
rec(
isGraph := true,
order := 1,
group := Group( [ () ] ),
schreierVector := [ -1 ],
adjacencies := [ [ ] ],
representatives := [ 1 ],
isSimple := true,
names := [ [ 3 ] ] )
</pre>
<p>
<p>
<h2><a name="SECT013">6.13 CollapsedCompleteOrbitsGraph</a></h2>
<p><p>
<a name = "SSEC013.1"></a>
<li><code>CollapsedCompleteOrbitsGraph( </code><var>G</var><code>, </code><var>gamma</var><code> )</code>
<li><code>CollapsedCompleteOrbitsGraph( </code><var>G</var><code>, </code><var>gamma</var><code>, </code><var>N</var><code> )</code>
<p>
Given a subgroup <var>G</var> of the automorphism group of the simple graph
<var>gamma</var>, this function returns a graph isomorphic to <var>delta</var>, defined as
follows. The vertices of <var>delta</var> are those <var>G</var>-orbits of the vertices
of <var>gamma</var> on which complete subgraphs are induced in <var>gamma</var>, and <var>x</var>
is joined to <var>y</var> in <var>delta</var> if and only if <var>xnot=y</var> and the subgraph of
<var>gamma</var> induced on <var>xcupy</var> is a complete graph. The name of a vertex
<var>v</var> in the returned graph is a list (not necessarily ordered) of the
vertex-names of <var>gamma</var> for the vertices in the <var>G</var>-orbit corresponding
to <var>v</var>.
<p>
If the optional parameter <var>N</var> is given, then it is assumed to be a
subgroup of <var>Aut(<var>gamma</var>)</var> preserving the set of <var>G</var>-orbits of the
vertices of <var>gamma</var> (for example, the normalizer in <code></code><var>gamma</var><code>.group</code> of
<var>G</var>). This information can make the function more efficient.
<p>
<pre>
gap> G := Group( (1,2) );;
gap> gamma := NullGraph( SymmetricGroup(3) );;
gap> CollapsedCompleteOrbitsGraph( G, gamma );
rec(
isGraph := true,
order := 1,
group := Group( [ () ] ),
schreierVector := [ -1 ],
adjacencies := [ [ ] ],
representatives := [ 1 ],
names := [ [ 3 ] ],
isSimple := true )
gap> gamma := CompleteGraph( SymmetricGroup(3) );;
gap> CollapsedCompleteOrbitsGraph( G, gamma );
rec(
isGraph := true,
order := 2,
group := Group( [ () ] ),
schreierVector := [ -1, -2 ],
adjacencies := [ [ 2 ], [ 1 ] ],
representatives := [ 1, 2 ],
names := [ [ 1, 2 ], [ 3 ] ],
isSimple := true )
</pre>
<p>
<p>
<h2><a name="SECT014">6.14 NewGroupGraph</a></h2>
<p><p>
<a name = "SSEC014.1"></a>
<li><code>NewGroupGraph( </code><var>G</var><code>, </code><var>gamma</var><code> )</code>
<p>
This function returns a copy <var>delta</var> of <var>gamma</var>, except that the group
associated with <var>delta</var> is <var>G</var>, which is assumed to be a subgroup of
<var>Aut(<var>delta</var>)</var>.
<p>
Note that the results of some functions of a graph depend on the group
associated with that graph (which must always be a subgroup of the
automorphism group of the graph).
<p>
<pre>
gap> gamma := JohnsonGraph(4,2);;
gap> aut := AutGroupGraph(gamma);
Group([ (3,4), (2,3)(4,5), (1,2)(5,6) ])
gap> Size(gamma.group);
24
gap> Size(aut);
48
gap> delta := NewGroupGraph( aut, gamma );;
gap> Size(delta.group);
48
gap> IsIsomorphicGraph( gamma, delta );
true
</pre>
<p>
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<P>
<address>grape manual<br>June 2006
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