[GRAPE] 6 Functions to construct new graphs from old

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# 6 Functions to construct new graphs from old

### Sections

1. InducedSubgraph
2. DistanceSetInduced
3. DistanceGraph
4. ComplementGraph
5. PointGraph
6. EdgeGraph
7. SwitchedGraph
8. UnderlyingGraph
9. QuotientGraph
10. BipartiteDouble
11. GeodesicsGraph
12. CollapsedIndependentOrbitsGraph
13. CollapsedCompleteOrbitsGraph
14. NewGroupGraph

This chapter describes functions to construct new graphs from old ones.

## 6.1 InducedSubgraph

- `InducedSubgraph(` gamma`,` V`)`
- `InducedSubgraph(` gamma`,` V`,` G`)`

  This function returns the subgraph of gamma induced on the vertex list V (which must not contain repeated elements). If the optional third parameter G is given, then it is assumed that G fixes V setwise, and is a group of automorphisms of the induced subgraph when restricted to V. In that case, the image of G acting on V is the group associated with the induced subgraph. If no such G is given then the associated group is trivial. The name of vertex i in the induced subgraph is equal to the name of vertex ``V`[`i`]` in gamma.

      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 ] ]

  ## 6.2 DistanceSetInduced

- `DistanceSetInduced(` gamma`,` distances`,` V`)`
- `DistanceSetInduced(` gamma`,` distances`,` V`,` G`)`

  Let V be a vertex or a nonempty list of vertices of gamma. This function returns the subgraph of gamma induced on the set of vertices w of gamma such that d(V ,w) is in distances (a list or singleton distance).

  The optional parameter G, if present, is assumed to be a subgroup of \Aut(gamma ) fixing V setwise. Including such a G can speed up the function.

  See also Distance and DistanceSet.

      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 ] ] )

  ## 6.3 DistanceGraph

- `DistanceGraph(` gamma`,` distances`)`

  This function returns the graph delta, with the same vertex-set (and vertex-names) as gamma, such that [x,y] is an edge of delta if and only if d(x,y) (in gamma) is in distances (a list or singleton distance).

      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 ] ]

  ## 6.4 ComplementGraph

- `ComplementGraph(` gamma`)`
- `ComplementGraph(` gamma`,` comploops`)`

  This function returns the complement of the graph gamma. The optional boolean parameter comploops determines whether or not loops/nonloops are complemented (default: `false` (loops/nonloops are not complemented)). The returned graph will have the same vertex-names as gamma.

      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

  ## 6.5 PointGraph

- `PointGraph(` gamma`)`
- `PointGraph(` gamma`,` v`)`

  Assuming that gamma is simple, connected, and bipartite, this function returns the induced subgraph on the connected component of `DistanceGraph(`gamma`,2)` containing the vertex v (default: v =1).

  Thus, if gamma is the incidence graph of a connected geometry of rank 2, and v represents a point, then the point graph of the geometry is returned.

      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

  ## 6.6 EdgeGraph

- `EdgeGraph(` gamma`)`

  This function return a graph isomorphic to the the edge graph (also called the line graph) of the simple graph gamma.

  This edge graph delta has the unordered edges of gamma as vertices, and e is joined to f in delta precisely when |eÇf|=1. The name of the vertex of the returned graph corresponding to the unordered edge [v,w] of gamma (with v < w) is `[VertexName(`gamma`,`v`),VertexName(`gamma`,`w`)]`.

      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 ] ]

  ## 6.7 SwitchedGraph

- `SwitchedGraph(` gamma`,` V`)`
- `SwitchedGraph(` gamma`,` V`,` H`)`

  This function returns the switched graph delta of the graph gamma, switched with respect to the vertex list (or singleton vertex) V.

  The returned graph delta has vertex-set (and vertex-names) the same as gamma. If vertices x,y of delta are both in V or both not in V, then [x,y] is an edge of delta if and only if [x,y] is an edge of gamma; otherwise [x,y] is an edge of delta if and only if [x,y] is not an edge of gamma. If the optional third argument H is given, then it is assumed to be a subgroup of Aut(gamma) stabilizing V setwise.

      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 ] ]

  ## 6.8 UnderlyingGraph

- `UnderlyingGraph(` gamma`)`

  This function returns the underlying graph delta of gamma. The graph delta has the same vertex-set (and vertex-names) as gamma, and has an edge [x,y] precisely when gamma has an edge [x,y] or an edge [y,x]. This function also sets the `isSimple` components of gamma and delta.

      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 )

  ## 6.9 QuotientGraph

- `QuotientGraph(` gamma`,` R`)`

  Let S be the smallest ``gamma`.group`-invariant equivalence relation on the vertices of gamma, such that S contains the relation R (which should be a list of ordered pairs (length 2 lists) of vertices of gamma). Then this function returns a graph isomorphic to the quotient delta of the graph gamma, defined as follows. The vertices of delta are the equivalence classes of S, and [X,Y] is an edge of delta if and only if [x,y] is an edge of gamma for some x Î X, y Î Y. The name of a vertex v in the returned graph is a list (not necessarily ordered) of the vertex-names of gamma for the vertices in the equivalence class corresponding to v.

      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

  ## 6.10 BipartiteDouble

- `BipartiteDouble(` gamma`)`

  This function returns the bipartite double of the graph gamma, as defined in BCN89.

      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

  ## 6.11 GeodesicsGraph

- `GeodesicsGraph(` gamma`,` x`,` y`)`

  This function returns the the graph induced on the set of geodesics in gamma between the vertices x and y, but including neither x nor y. This function is only for a simple graph gamma.

      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 ] ]

  ## 6.12 CollapsedIndependentOrbitsGraph

- `CollapsedIndependentOrbitsGraph(` G`,` gamma`)`
- `CollapsedIndependentOrbitsGraph(` G`,` gamma`,` N`)`

  Given a subgroup G of the automorphism group of the simple graph gamma, this function returns a graph isomorphic to delta, defined as follows. The vertices of delta are those G-orbits of the vertices of gamma that are independent sets in gamma, and x is joined to y in delta if and only if xÈy is not an independent set in gamma. The name of a vertex v in the returned graph is a list (not necessarily ordered) of the vertex-names of gamma for the vertices in the G-orbit corresponding to v.

  If the optional parameter N is given, then it is assumed to be a subgroup of \Aut(gamma ) preserving the set of G-orbits of the vertices of gamma (for example, the normalizer in ``gamma`.group` of G). This information can make the function more efficient.

      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 ] ] )

  ## 6.13 CollapsedCompleteOrbitsGraph

- `CollapsedCompleteOrbitsGraph(` G`,` gamma`)`
- `CollapsedCompleteOrbitsGraph(` G`,` gamma`,` N`)`

  Given a subgroup G of the automorphism group of the simple graph gamma, this function returns a graph isomorphic to delta, defined as follows. The vertices of delta are those G-orbits of the vertices of gamma on which complete subgraphs are induced in gamma, and x is joined to y in delta if and only if x ¹ y and the subgraph of gamma induced on xÈy is a complete graph. The name of a vertex v in the returned graph is a list (not necessarily ordered) of the vertex-names of gamma for the vertices in the G-orbit corresponding to v.

  If the optional parameter N is given, then it is assumed to be a subgroup of \Aut(gamma ) preserving the set of G-orbits of the vertices of gamma (for example, the normalizer in ``gamma`.group` of G). This information can make the function more efficient.

      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 )

  ## 6.14 NewGroupGraph

- `NewGroupGraph(` G`,` gamma`)`

  This function returns a copy delta of gamma, except that the group associated with delta is G, which is assumed to be a subgroup of \Aut(delta ).

  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).

      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

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