Incidence structures (i.e. hypergraphs, i.e. set systems)

An incidence structure is specified by a list of points, blocks, or an incidence matrix ([1], [2]). IncidenceStructure instances have the following methods:

ground_set() Return the ground set (i.e the list of points).
num_points() Return the size of the ground set.
num_blocks() Return the number of blocks.
blocks() Return the list of blocks.
block_sizes() Return the set of block sizes.
degree() Return the degree of a point p
is_connected() Test whether the design is connected.
is_simple() Test whether this design is simple (i.e. no repeated block).
incidence_matrix() Return the incidence matrix \(A\) of the design
incidence_graph() Return the incidence graph of the design
packing() Return a maximum packing
is_t_design() Test whether self is a \(t-(v,k,l)\) design.
dual() Return the dual design.
automorphism_group() Return the automorphism group
edge_coloring() Return an optimal edge coloring`

REFERENCES:

[1]Block designs and incidence structures from wikipedia, Wikipedia article Block_design Wikipedia article Incidence_structure
[2]
  1. Assmus, J. Key, Designs and their codes, CUP, 1992.

AUTHORS:

  • Peter Dobcsanyi and David Joyner (2007-2008)

    This is a significantly modified form of part of the module block_design.py (version 0.6) written by Peter Dobcsanyi peter@designtheory.org.

  • Vincent Delecroix (2014): major rewrite

Methods

class sage.combinat.designs.incidence_structures.IncidenceStructure(points=None, blocks=None, incidence_matrix=None, name=None, check=True, test=None, copy=True)

Bases: object

A base class for incidence structures (i.e. hypergraphs, i.e. set systems)

An incidence structure (i.e. hypergraph, i.e. set system) can be defined from a collection of blocks (i.e. sets, i.e. edges), optionally with an explicit ground set (i.e. point set, i.e. vertex set). Alternatively they can be defined from a binary incidence matrix.

INPUT:

  • points – (i.e. ground set, i.e. vertex set) the underlying set. If points is an integer \(v\), then the set is considered to be \(\{0, ..., v-1\}\).

    Note

    The following syntax, where points is ommitted, automatically defines the ground set as the union of the blocks:

    sage: H = IncidenceStructure([['a','b','c'],['c','d','e']])
    sage: H.ground_set()
    ['a', 'b', 'c', 'd', 'e']
    
  • blocks – (i.e. edges, i.e. sets) the blocks defining the incidence structure. Can be any iterable.

  • incidence_matrix – a binary incidence matrix. Each column represents a set.

  • name (a string, such as “Fano plane”).

  • check – whether to check the input

  • copy – (use with caution) if set to False then blocks must be a list of lists of integers. The list will not be copied but will be modified in place (each block is sorted, and the whole list is sorted). Your blocks object will become the IncidenceStructure instance’s internal data.

EXAMPLES:

An incidence structure can be constructed by giving the number of points and the list of blocks:

sage: designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
Incidence structure with 7 points and 7 blocks

Only providing the set of blocks is sufficient. In this case, the ground set is defined as the union of the blocks:

sage: IncidenceStructure([[1,2,3],[2,3,4]])
Incidence structure with 4 points and 2 blocks

Or by its adjacency matrix (a \(\{0,1\}\)-matrix in which rows are indexed by points and columns by blocks):

sage: m = matrix([[0,1,0],[0,0,1],[1,0,1],[1,1,1]])
sage: designs.IncidenceStructure(m)
Incidence structure with 4 points and 3 blocks

The points can be any (hashable) object:

sage: V = [(0,'a'),(0,'b'),(1,'a'),(1,'b')]
sage: B = [(V[0],V[1],V[2]), (V[1],V[2]), (V[0],V[2])]
sage: I = designs.IncidenceStructure(V, B)
sage: I.ground_set()
[(0, 'a'), (0, 'b'), (1, 'a'), (1, 'b')]
sage: I.blocks()
[[(0, 'a'), (0, 'b'), (1, 'a')], [(0, 'a'), (1, 'a')], [(0, 'b'), (1, 'a')]]

The order of the points and blocks does not matter as they are sorted on input (see trac ticket #11333):

sage: A = designs.IncidenceStructure([0,1,2], [[0],[0,2]])
sage: B = designs.IncidenceStructure([1,0,2], [[0],[2,0]])
sage: B == A
True

sage: C = designs.BlockDesign(2, [[0], [1,0]])
sage: D = designs.BlockDesign(2, [[0,1], [0]])
sage: C == D
True

If you care for speed, you can set copy to False, but in that case, your input must be a list of lists and the ground set must be \({0, ..., v-1}\):

sage: blocks = [[0,1],[2,0],[1,2]]  # a list of lists of integers
sage: I = designs.IncidenceStructure(3, blocks, copy=False)
sage: I.blocks(copy=False) is blocks
True
automorphism_group()

Return the subgroup of the automorphism group of the incidence graph which respects the P B partition. It is (isomorphic to) the automorphism group of the block design, although the degrees differ.

EXAMPLES:

sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
Incidence structure with 7 points and 7 blocks
sage: G = P.automorphism_group()
sage: G.is_isomorphic(PGL(3,2))
True
sage: G
Permutation Group with generators [(2,3)(4,5), (2,4)(3,5), (1,2)(4,6), (0,1)(4,5)]

A non self-dual example:

sage: IS = designs.IncidenceStructure(range(4), [[0,1,2,3],[1,2,3]])
sage: IS.automorphism_group().cardinality()
6
sage: IS.dual().automorphism_group().cardinality()
1

Examples with non-integer points:

sage: I = designs.IncidenceStructure('abc', ('ab','ac','bc'))
sage: I.automorphism_group()
Permutation Group with generators [('b','c'), ('a','b')]
sage: designs.IncidenceStructure([[(1,2),(3,4)]]).automorphism_group()
Permutation Group with generators [((1,2),(3,4))]
block_design_checker(t, v, k, lmbda, type=None)

This method is deprecated and will soon be removed (see trac ticket #16553). You could use is_t_design() instead.

This is not a wrapper for GAP Design’s IsBlockDesign. The GAP Design function IsBlockDesign http://www.gap-system.org/Manuals/pkg/design/htm/CHAP004.htm apparently simply checks the record structure and no mathematical properties. Instead, the function below checks some necessary (but not sufficient) “easy” identities arising from the identity.

INPUT:

  • t - the t as in “t-design”
  • v - the number of points
  • k - the number of blocks incident to a point
  • lmbda - each t-tuple of points should be incident with lmbda blocks
  • type - can be ‘simple’ or ‘binary’ or ‘connected’ Depending on the option, this wraps IsBinaryBlockDesign, IsSimpleBlockDesign, or IsConnectedBlockDesign.
    • Binary: no block has a repeated element.
    • Simple: no block is repeated.
    • Connected: its incidence graph is a connected graph.

WARNING: This is very fast but can return false positives.

EXAMPLES:

sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.is_t_design(return_parameters=True)
(True, (2, 7, 3, 1))
sage: BD.block_design_checker(2, 7, 3, 1)
doctest:...: DeprecationWarning: .block_design_checker(v,t,k,lmbda) is deprecated; please use
.is_t_design(v,t,k,lmbda) instead
See http://trac.sagemath.org/16553 for details.
True

sage: BD.block_design_checker(2, 7, 3, 1,"binary")
doctest:...: DeprecationWarning: .block_design_checker(type='binary') is
deprecated; use .is_binary() instead
See http://trac.sagemath.org/16553 for details.
True

sage: BD.block_design_checker(2, 7, 3, 1,"connected")
doctest:...: DeprecationWarning: block_design_checker(type='connected') is
deprecated, please use .is_connected() instead
See http://trac.sagemath.org/16553 for details.
True

sage: BD.block_design_checker(2, 7, 3, 1,"simple")
doctest:...: DeprecationWarning: .block_design_checker(type='simple')
is deprecated; all designs here are simple!
See http://trac.sagemath.org/16553 for details.
True
block_sizes()

Return the set of block sizes.

EXAMPLES:

sage: BD = designs.IncidenceStructure(8, [[0,1,3],[1,4,5,6],[1,2],[5,6,7]])
sage: BD.block_sizes()
[3, 2, 4, 3]
sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.block_sizes()
[3, 3, 3, 3, 3, 3, 3]
blocks(copy=True)

Return the list of blocks.

INPUT:

  • copy (boolean) – True by default. When set to False, a pointer toward the object’s internal data is given. Set it to False only if you know what you are doing.

EXAMPLES:

sage: BD = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.blocks()
[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]

What you should pay attention to:

sage: blocks = BD.blocks(copy=False)
sage: del blocks[0:6]
sage: BD
Incidence structure with 7 points and 1 blocks
degree(p=None)

Return the degree of a point p

The degree of a point \(p\) is the number of blocks that contain it.

INPUT:

  • p – a point. If set to None (default), a dictionary associating the points with their degrees is returned.

EXAMPLES:

sage: designs.steiner_triple_system(9).degree(3)
4
sage: designs.steiner_triple_system(9).degree()
{0: 4, 1: 4, 2: 4, 3: 4, 4: 4, 5: 4, 6: 4, 7: 4, 8: 4}
dual(algorithm=None)

Return the dual of the incidence structure.

INPUT:

  • algorithm – whether to use Sage’s implementation (algorithm=None, default) or use GAP’s (algorithm="gap").

    Note

    The algorithm="gap" option requires GAP’s Design package (included in the gap_packages Sage spkg).

EXAMPLES:

The dual of a projective plane is a projective plane:

sage: PP = designs.DesarguesianProjectivePlaneDesign(4)
sage: PP.dual().is_t_design(return_parameters=True)
(True, (2, 21, 5, 1))

TESTS:

sage: D = designs.IncidenceStructure(4, [[0,2],[1,2,3],[2,3]])
sage: D
Incidence structure with 4 points and 3 blocks
sage: D.dual()
Incidence structure with 3 points and 4 blocks
sage: print D.dual(algorithm="gap")       # optional - gap_packages
IncidenceStructure<points=[0, 1, 2], blocks=[[0], [0, 1, 2], [1], [1, 2]]>
sage: blocks = [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]]
sage: BD = designs.IncidenceStructure(7, blocks, name="FanoPlane");
sage: BD
Incidence structure with 7 points and 7 blocks
sage: print BD.dual(algorithm="gap")         # optional - gap_packages
IncidenceStructure<points=[0, 1, 2, 3, 4, 5, 6], blocks=[[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]]>
sage: BD.dual()
Incidence structure with 7 points and 7 blocks

REFERENCE:

dual_design(*args, **kwds)

Deprecated: Use dual() instead. See trac ticket #16553 for details.

dual_incidence_structure(*args, **kwds)

Deprecated: Use dual() instead. See trac ticket #16553 for details.

edge_coloring()

Compute a proper edge-coloring.

A proper edge-coloring is an assignment of colors to the sets of the incidence structure such that two sets with non-empty intersection receive different colors. The coloring returned minimizes the number of colors.

OUTPUT:

A partition of the sets into color classes.

EXAMPLES:

sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Incidence structure with 6 points and 4 blocks
sage: C = H.edge_coloring()
sage: C # random
[[[3, 4, 5]], [[2, 3, 4]], [[4, 5, 6], [1, 2, 3]]]
sage: Set(map(Set,sum(C,[]))) == Set(map(Set,H.blocks()))
True
ground_set(copy=True)

Return the ground set (i.e the list of points).

INPUT:

  • copy (boolean) – True by default. When set to False, a pointer toward the object’s internal data is given. Set it to False only if you know what you are doing.

EXAMPLES:

sage: designs.IncidenceStructure(3, [[0,1],[0,2]]).ground_set()
[0, 1, 2]
incidence_graph()

Return the incidence graph of the design, where the incidence matrix of the design is the adjacency matrix of the graph.

EXAMPLE:

sage: BD = designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.incidence_graph()
Bipartite graph on 14 vertices
sage: A = BD.incidence_matrix()
sage: Graph(block_matrix([[A*0,A],[A.transpose(),A*0]])) == BD.incidence_graph()
True

REFERENCE:

  • Sage Reference Manual on Graphs
incidence_matrix()

Return the incidence matrix \(A\) of the design. A is a \((v \times b)\) matrix defined by: A[i,j] = 1 if i is in block B_j and 0 otherwise.

EXAMPLES:

sage: BD = designs.IncidenceStructure(7, [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.block_sizes()
[3, 3, 3, 3, 3, 3, 3]
sage: BD.incidence_matrix()
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[1 0 0 0 0 1 1]
[0 1 0 1 0 1 0]
[0 1 0 0 1 0 1]
[0 0 1 1 0 0 1]
[0 0 1 0 1 1 0]

sage: I = designs.IncidenceStructure('abc', ('ab','abc','ac','c'))
sage: I.incidence_matrix()
[1 1 1 0]
[1 1 0 0]
[0 1 1 1]
is_block_design(*args, **kwds)

Deprecated: Use is_t_design() instead. See trac ticket #16553 for details.

is_connected()

Test whether the design is connected.

EXAMPLES:

sage: designs.IncidenceStructure(3, [[0,1],[0,2]]).is_connected()
True
sage: designs.IncidenceStructure(4, [[0,1],[2,3]]).is_connected()
False
is_simple()

Test whether this design is simple (i.e. no repeated block).

EXAMPLES:

sage: designs.IncidenceStructure(3, [[0,1],[1,2],[0,2]]).is_simple()
True
sage: designs.IncidenceStructure(3, [[0],[0]]).is_simple()
False

sage: V = [(0,'a'),(0,'b'),(1,'a'),(1,'b')]
sage: B = [[V[0],V[1]], [V[1],V[2]]]
sage: I = designs.IncidenceStructure(V, B)
sage: I.is_simple()
True
sage: I2 = designs.IncidenceStructure(V, B*2)
sage: I2.is_simple()
False
is_t_design(t=None, v=None, k=None, l=None, return_parameters=False)

Test whether self is a \(t-(v,k,l)\) design.

A \(t-(v,k,\lambda)\) (sometimes called \(t\)-design for short) is a block design in which:

  • the underlying set has cardinality \(v\)
  • the blocks have size \(k\)
  • each \(t\)-subset of points is covered by \(\lambda\) blocks

INPUT:

  • t,v,k,l (integers) – their value is set to None by default. The function tests whether the design is a t-(v,k,l) design using the provided values and guesses the others. Note that \(l`\) cannot be specified if t is not.
  • return_parameters (boolean)– whether to return the parameters of the \(t\)-design. If set to True, the function returns a pair (boolean_answer,(t,v,k,l)).

EXAMPLES:

sage: fano_blocks = [[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]]
sage: BD = designs.IncidenceStructure(7, fano_blocks)
sage: BD.is_t_design()
True
sage: BD.is_t_design(return_parameters=True)
(True, (2, 7, 3, 1))
sage: BD.is_t_design(2, 7, 3, 1)
True
sage: BD.is_t_design(1, 7, 3, 3)
True
sage: BD.is_t_design(0, 7, 3, 7)
True

sage: BD.is_t_design(0,6,3,7) or BD.is_t_design(0,7,4,7) or BD.is_t_design(0,7,3,8)
False

sage: BD = designs.AffineGeometryDesign(3, 1, GF(2))
sage: BD.is_t_design(1)
True
sage: BD.is_t_design(2)
True

Steiner triple and quadruple systems are other names for \(2-(v,3,1)\) and \(3-(v,4,1)\) designs:

sage: S3_9 = designs.steiner_triple_system(9)
sage: S3_9.is_t_design(2,9,3,1)
True

sage: blocks = designs.steiner_quadruple_system(8)
sage: S4_8 = designs.IncidenceStructure(8, blocks)
sage: S4_8.is_t_design(3,8,4,1)
True

sage: blocks = designs.steiner_quadruple_system(14)
sage: S4_14 = designs.IncidenceStructure(14, blocks)
sage: S4_14.is_t_design(3,14,4,1)
True

Some examples of Witt designs that need the gap database:

sage: BD = designs.WittDesign(9)         # optional - gap_packages
sage: BD.is_t_design(2,9,3,1)            # optional - gap_packages
True
sage: W12 = designs.WittDesign(12)       # optional - gap_packages
sage: W12.is_t_design(5,12,6,1)          # optional - gap_packages
True
sage: W12.is_t_design(4)                 # optional - gap_packages
True

Further examples:

sage: D = designs.IncidenceStructure(4,[[],[]])
sage: D.is_t_design(return_parameters=True)
(True,  (0, 4, 0, 2))

sage: D = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3]])
sage: D.is_t_design(return_parameters=True)
(True, (0, 4, 2, 3))

sage: D = designs.IncidenceStructure(4, [[0],[1],[2],[3]])
sage: D.is_t_design(return_parameters=True)
(True, (1, 4, 1, 1))

sage: D = designs.IncidenceStructure(4,[[0,1],[2,3]])
sage: D.is_t_design(return_parameters=True)
(True, (1, 4, 2, 1))

sage: D = designs.IncidenceStructure(4, [range(4)])
sage: D.is_t_design(return_parameters=True)
(True, (4, 4, 4, 1))

TESTS:

sage: blocks = designs.steiner_quadruple_system(8)
sage: S4_8 = designs.IncidenceStructure(8, blocks)
sage: R = range(15)
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(3,v,k,l)]
[(8, 4, 1)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(2,v,k,l)]
[(8, 4, 3)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(1,v,k,l)]
[(8, 4, 7)]
sage: [(v,k,l) for v in R for k in R for l in R if S4_8.is_t_design(0,v,k,l)]
[(8, 4, 14)]
sage: A = designs.AffineGeometryDesign(3, 1, GF(2))
sage: A.is_t_design(return_parameters=True)
(True, (2, 8, 2, 1))
sage: A = designs.AffineGeometryDesign(4, 2, GF(2))
sage: A.is_t_design(return_parameters=True)
(True, (3, 16, 4, 1))
sage: I = designs.IncidenceStructure(2, [])
sage: I.is_t_design(return_parameters=True)
(True, (0, 2, 0, 0))
sage: I = designs.IncidenceStructure(2, [[0],[0,1]])
sage: I.is_t_design(return_parameters=True)
(False, (0, 0, 0, 0))
num_blocks()

Return the number of blocks.

EXAMPLES:

sage: designs.DesarguesianProjectivePlaneDesign(2).num_blocks()
7
sage: B = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]])
sage: B.num_blocks()
5
num_points()

Return the size of the ground set.

EXAMPLES:

sage: designs.DesarguesianProjectivePlaneDesign(2).num_points()
7
sage: B = designs.IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]])
sage: B.num_points()
4
packing(solver=None, verbose=0)

Return a maximum packing

A maximum packing in a hypergraph is collection of disjoint sets/blocks of maximal cardinality. This problem is NP-complete in general, and in particular on 3-uniform hypergraphs. It is solved here with an Integer Linear Program.

For more information, see the Wikipedia article Packing_in_a_hypergraph.

INPUT:

  • solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
  • verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet. Only useful when algorithm == "LP".

EXAMPLE:

sage; IncidenceStructure([[1,2],[3,"A"],[2,3]]).packing()
[[1, 2], [3, 'A']]
sage: len(designs.steiner_triple_system(9).packing())
3
parameters()

Deprecated function. You should use is_t_design() instead.

EXAMPLES:

sage: I = designs.IncidenceStructure('abc', ['ab','ac','bc'])
sage: I.parameters()
doctest:...: DeprecationWarning: .parameters() is deprecated. Use
`is_t_design` instead
See http://trac.sagemath.org/16553 for details.
(2, 3, 2, 1)
points(*args, **kwds)

Deprecated: Use ground_set() instead. See trac ticket #16553 for details.

sage.combinat.designs.incidence_structures.IncidenceStructureFromMatrix(M, name=None)

Deprecated function that builds an incidence structure from a matrix.

You should now use designs.IncidenceStructure(incidence_matrix=M).

INPUT:

  • M – a binary matrix. Creates a set of “points” from the rows and a set of “blocks” from the columns.

EXAMPLES:

sage: BD1 = designs.IncidenceStructure(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: M = BD1.incidence_matrix()
sage: BD2 = IncidenceStructureFromMatrix(M)
doctest:...: DeprecationWarning: IncidenceStructureFromMatrix is deprecated.
Please use designs.IncidenceStructure(incidence_matrix=M) instead.
See http://trac.sagemath.org/16553 for details.
sage: BD1 == BD2
True

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