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:
| 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. | |
| Return the set of block sizes. | |
| Return the list of blocks. | |
| Return a canonical label for the incidence structure. | |
| Compute a (weak) \(k\)-coloring of the hypergraph. | |
| Return the complement of the incidence structure. | |
| Return a copy of the incidence structure. | |
| Return the degree of a point  | |
| Return the degree of all sets of given size, or the degree of all points. | |
| Return the dual of the incidence structure. | |
| Compute a proper edge-coloring. | |
| Return the ground set (i.e the list of points). | |
| Return the incidence graph of the incidence structure. | |
| Return the incidence matrix \(A\) of the design. A is a \((v \times b)\) matrix defined by:  | |
| Return the substructure induced by a set of points. | |
| Return the intersection graph of the incidence structure. | |
| Check whether  | |
| Test whether the design is connected. | |
| Test if the incidence structure is a generalized quadrangle. | |
| Return whether the two incidence structures are isomorphic. | |
| Test whether the incidence structure is \(r\)-regular. | |
| Test whether the hypergraph is resolvable. | |
| Test whether this design is simple (i.e. no repeated block). | |
| Check whether the input is a spread for  | |
| Test whether  | |
| Test whether the incidence structure is \(k\)-uniform | |
| Iterate over all copies of  | |
| Return the number of blocks. | |
| Return the size of the ground set. | |
| Return a maximum packing. | |
| Return the rank of the hypergraph (the maximum size of a block). | |
| Relabel the ground set. | |
| Return the trace of a set of points. | 
REFERENCES:
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, copy=True)[source]¶
- 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- pointsis an integer \(v\), then the set is considered to be \(\{0, ..., v-1\}\).- Note - The following syntax, where - pointsis omitted, automatically defines the ground set as the union of the blocks:- sage: H = IncidenceStructure([['a','b','c'],['c','d','e']]) sage: sorted(H.ground_set()) ['a', 'b', 'c', 'd', 'e'] - >>> from sage.all import * >>> H = IncidenceStructure([['a','b','c'],['c','d','e']]) >>> sorted(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– string (such as “Fano plane”)
- check– whether to check the input
- copy– (use with caution) if set to- Falsethen- blocksmust 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- blocksobject will become the- IncidenceStructureinstance’s internal data.
 - EXAMPLES: - An incidence structure can be constructed by giving the number of points and the list of blocks: - sage: 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 - >>> from sage.all import * >>> IncidenceStructure(Integer(7), [[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)],[Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(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 - >>> from sage.all import * >>> IncidenceStructure([[Integer(1),Integer(2),Integer(3)],[Integer(2),Integer(3),Integer(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]]) # needs sage.modules sage: IncidenceStructure(m) # needs sage.modules Incidence structure with 4 points and 3 blocks - >>> from sage.all import * >>> m = matrix([[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)],[Integer(1),Integer(0),Integer(1)],[Integer(1),Integer(1),Integer(1)]]) # needs sage.modules >>> IncidenceStructure(m) # needs sage.modules 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 = 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')]] - >>> from sage.all import * >>> V = [(Integer(0),'a'),(Integer(0),'b'),(Integer(1),'a'),(Integer(1),'b')] >>> B = [(V[Integer(0)],V[Integer(1)],V[Integer(2)]), (V[Integer(1)],V[Integer(2)]), (V[Integer(0)],V[Integer(2)])] >>> I = IncidenceStructure(V, B) >>> I.ground_set() [(0, 'a'), (0, 'b'), (1, 'a'), (1, 'b')] >>> 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 Issue #11333): - sage: A = IncidenceStructure([0,1,2], [[0],[0,2]]) sage: B = IncidenceStructure([1,0,2], [[0],[2,0]]) sage: B == A True sage: C = BlockDesign(2, [[0], [1,0]]) sage: D = BlockDesign(2, [[0,1], [0]]) sage: C == D True - >>> from sage.all import * >>> A = IncidenceStructure([Integer(0),Integer(1),Integer(2)], [[Integer(0)],[Integer(0),Integer(2)]]) >>> B = IncidenceStructure([Integer(1),Integer(0),Integer(2)], [[Integer(0)],[Integer(2),Integer(0)]]) >>> B == A True >>> C = BlockDesign(Integer(2), [[Integer(0)], [Integer(1),Integer(0)]]) >>> D = BlockDesign(Integer(2), [[Integer(0),Integer(1)], [Integer(0)]]) >>> C == D True - If you care for speed, you can set - copyto- 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 = IncidenceStructure(3, blocks, copy=False) sage: I._blocks is blocks True - >>> from sage.all import * >>> blocks = [[Integer(0),Integer(1)],[Integer(2),Integer(0)],[Integer(1),Integer(2)]] # a list of lists of integers >>> I = IncidenceStructure(Integer(3), blocks, copy=False) >>> I._blocks is blocks True - automorphism_group()[source]¶
- 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: # needs sage.groups sage.rings.finite_rings sage: P = designs.DesarguesianProjectivePlaneDesign(2); P (7,3,1)-Balanced Incomplete Block Design sage: G = P.automorphism_group() sage: G.is_isomorphic(PGL(3,2)) True sage: G Permutation Group with generators [...] sage: G.cardinality() 168 - >>> from sage.all import * >>> # needs sage.groups sage.rings.finite_rings >>> P = designs.DesarguesianProjectivePlaneDesign(Integer(2)); P (7,3,1)-Balanced Incomplete Block Design >>> G = P.automorphism_group() >>> G.is_isomorphic(PGL(Integer(3),Integer(2))) True >>> G Permutation Group with generators [...] >>> G.cardinality() 168 - A non self-dual example: - sage: IS = IncidenceStructure(list(range(4)), [[0,1,2,3],[1,2,3]]) sage: IS.automorphism_group().cardinality() # needs sage.groups 6 sage: IS.dual().automorphism_group().cardinality() # needs sage.groups sage.modules 1 - >>> from sage.all import * >>> IS = IncidenceStructure(list(range(Integer(4))), [[Integer(0),Integer(1),Integer(2),Integer(3)],[Integer(1),Integer(2),Integer(3)]]) >>> IS.automorphism_group().cardinality() # needs sage.groups 6 >>> IS.dual().automorphism_group().cardinality() # needs sage.groups sage.modules 1 - Examples with non-integer points: - sage: I = IncidenceStructure('abc', ('ab','ac','bc')) sage: I.automorphism_group() # needs sage.groups Permutation Group with generators [('b','c'), ('a','b')] sage: IncidenceStructure([[(1,2),(3,4)]]).automorphism_group() # needs sage.groups Permutation Group with generators [((1,2),(3,4))] - >>> from sage.all import * >>> I = IncidenceStructure('abc', ('ab','ac','bc')) >>> I.automorphism_group() # needs sage.groups Permutation Group with generators [('b','c'), ('a','b')] >>> IncidenceStructure([[(Integer(1),Integer(2)),(Integer(3),Integer(4))]]).automorphism_group() # needs sage.groups Permutation Group with generators [((1,2),(3,4))] 
 - block_sizes()[source]¶
- Return the set of block sizes. - EXAMPLES: - sage: BD = IncidenceStructure(8, [[0,1,3],[1,4,5,6],[1,2],[5,6,7]]) sage: BD.block_sizes() [3, 2, 4, 3] sage: BD = 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] - >>> from sage.all import * >>> BD = IncidenceStructure(Integer(8), [[Integer(0),Integer(1),Integer(3)],[Integer(1),Integer(4),Integer(5),Integer(6)],[Integer(1),Integer(2)],[Integer(5),Integer(6),Integer(7)]]) >>> BD.block_sizes() [3, 2, 4, 3] >>> BD = IncidenceStructure(Integer(7),[[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)],[Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(5)]]) >>> BD.block_sizes() [3, 3, 3, 3, 3, 3, 3] 
 - blocks()[source]¶
- Return the list of blocks. - EXAMPLES: - sage: BD = 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]] - >>> from sage.all import * >>> BD = IncidenceStructure(Integer(7),[[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)],[Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(5)]]) >>> BD.blocks() [[0, 1, 2], [0, 3, 4], [0, 5, 6], [1, 3, 5], [1, 4, 6], [2, 3, 6], [2, 4, 5]] 
 - canonical_label()[source]¶
- Return a canonical label for the incidence structure. - A canonical label is relabeling of the points into integers \(\{0,...,n-1\}\) such that isomorphic incidence structures are relabelled to equal objects. - EXAMPLES: - sage: # needs sage.schemes sage: fano1 = designs.balanced_incomplete_block_design(7,3) sage: fano2 = designs.projective_plane(2) sage: fano1 == fano2 False sage: fano1.relabel(fano1.canonical_label()) sage: fano2.relabel(fano2.canonical_label()) sage: fano1 == fano2 True - >>> from sage.all import * >>> # needs sage.schemes >>> fano1 = designs.balanced_incomplete_block_design(Integer(7),Integer(3)) >>> fano2 = designs.projective_plane(Integer(2)) >>> fano1 == fano2 False >>> fano1.relabel(fano1.canonical_label()) >>> fano2.relabel(fano2.canonical_label()) >>> fano1 == fano2 True 
 - coloring(k, solver=None, verbose=None, integrality_tolerance=0)[source]¶
- Compute a (weak) \(k\)-coloring of the hypergraph. - A weak coloring of a hypergraph \(\mathcal H\) is an assignment of colors to its vertices such that no set is monochromatic. - INPUT: - k– integer; compute a coloring with \(k\) colors if an integer is provided, otherwise returns an optimal coloring (i.e. with the minimum possible number of colors).
- solver– (default:- None) specify a Mixed Integer Linear Programming (MILP) solver to be used. If set to- None, the default one is used. For more information on MILP solvers and which default solver is used, see the method- solveof the class- MixedIntegerLinearProgram.
- verbose– nonnegative integer (default: \(0\)); set the level of verbosity you want from the linear program solver. Since the problem is \(NP\)-complete, its solving may take some time depending on the graph. A value of \(0\) means that there will be no message printed by the solver.
- integrality_tolerance– parameter for use with MILP solvers over an inexact base ring; see- MixedIntegerLinearProgram.get_values()
 - EXAMPLES: - The Fano plane has chromatic number 3: - sage: len(designs.steiner_triple_system(7).coloring()) # needs sage.numerical.mip 3 - >>> from sage.all import * >>> len(designs.steiner_triple_system(Integer(7)).coloring()) # needs sage.numerical.mip 3 - One admissible 3-coloring: - sage: designs.steiner_triple_system(7).coloring() # not tested # needs sage.numerical.mip [[0, 2, 5, 1], [4, 3], [6]] - >>> from sage.all import * >>> designs.steiner_triple_system(Integer(7)).coloring() # not tested # needs sage.numerical.mip [[0, 2, 5, 1], [4, 3], [6]] - The chromatic number of a graph is equal to the chromatic number of its 2-uniform corresponding hypergraph: - sage: g = graphs.PetersenGraph() sage: H = IncidenceStructure(g.edges(sort=True, labels=False)) sage: len(g.coloring()) 3 sage: len(H.coloring()) # needs sage.numerical.mip 3 - >>> from sage.all import * >>> g = graphs.PetersenGraph() >>> H = IncidenceStructure(g.edges(sort=True, labels=False)) >>> len(g.coloring()) 3 >>> len(H.coloring()) # needs sage.numerical.mip 3 
 - complement(uniform=False)[source]¶
- Return the complement of the incidence structure. - Two different definitions of “complement” are made available, according to the value of - uniform.- INPUT: - uniform– boolean- if set to - False(default), returns the incidence structure whose blocks are the complements of all blocks of the incidence structure.
- If set to - Trueand the incidence structure is \(k\)-uniform, returns the incidence structure whose blocks are all \(k\)-sets of the ground set that do not appear in- self.
 
 - EXAMPLES: - The complement of a - BalancedIncompleteBlockDesignis also a \(2\)-design:- sage: bibd = designs.balanced_incomplete_block_design(13,4) # needs sage.schemes sage: bibd.is_t_design(return_parameters=True) # needs sage.schemes (True, (2, 13, 4, 1)) sage: bibd.complement().is_t_design(return_parameters=True) # needs sage.schemes (True, (2, 13, 9, 6)) - >>> from sage.all import * >>> bibd = designs.balanced_incomplete_block_design(Integer(13),Integer(4)) # needs sage.schemes >>> bibd.is_t_design(return_parameters=True) # needs sage.schemes (True, (2, 13, 4, 1)) >>> bibd.complement().is_t_design(return_parameters=True) # needs sage.schemes (True, (2, 13, 9, 6)) - The “uniform” complement of a graph is a graph: - sage: g = graphs.PetersenGraph() sage: G = IncidenceStructure(g.edges(sort=True, labels=False)) sage: H = G.complement(uniform=True) sage: h = Graph(H.blocks()) sage: g == h False sage: g == h.complement() True - >>> from sage.all import * >>> g = graphs.PetersenGraph() >>> G = IncidenceStructure(g.edges(sort=True, labels=False)) >>> H = G.complement(uniform=True) >>> h = Graph(H.blocks()) >>> g == h False >>> g == h.complement() True 
 - copy()[source]¶
- Return a copy of the incidence structure. - EXAMPLES: - sage: IS = IncidenceStructure([[1,2,3,"e"]], name='Test') sage: IS Incidence structure with 4 points and 1 blocks sage: copy(IS) Incidence structure with 4 points and 1 blocks sage: [1, 2, 3, 'e'] in copy(IS) True sage: copy(IS)._name 'Test' - >>> from sage.all import * >>> IS = IncidenceStructure([[Integer(1),Integer(2),Integer(3),"e"]], name='Test') >>> IS Incidence structure with 4 points and 1 blocks >>> copy(IS) Incidence structure with 4 points and 1 blocks >>> [Integer(1), Integer(2), Integer(3), 'e'] in copy(IS) True >>> copy(IS)._name 'Test' 
 - degree(p=None, subset=False)[source]¶
- Return the degree of a point - p(or a set of points).- The degree of a point (or set of points) is the number of blocks that contain it. - INPUT: - p– a point (or a set of points) of the incidence structure
- subset– boolean (default:- False); whether to interpret the argument as a set of point or as a point (default)
 - EXAMPLES: - sage: designs.steiner_triple_system(9).degree(3) 4 sage: designs.steiner_triple_system(9).degree({1,2},subset=True) 1 - >>> from sage.all import * >>> designs.steiner_triple_system(Integer(9)).degree(Integer(3)) 4 >>> designs.steiner_triple_system(Integer(9)).degree({Integer(1),Integer(2)},subset=True) 1 
 - degrees(size=None)[source]¶
- Return the degree of all sets of given size, or the degree of all points. - The degree of a point (or set of point) is the number of blocks that contain it. - INPUT: - size– integer; return the degree of all subsets of points of cardinality- size. When- size=None, the function outputs the degree of all points.- Note - When - size=Nonethe output is indexed by the points. When- size=1it is indexed by tuples of size 1. This is the same information, stored slightly differently.
 - OUTPUT: a dictionary whose values are degrees and keys are either: - the points of the incidence structure if - size=None(default)
- the subsets of size - sizeof the points stored as tuples
 - EXAMPLES: - sage: IncidenceStructure([[1,2,3],[1,4]]).degrees(2) {(1, 2): 1, (1, 3): 1, (1, 4): 1, (2, 3): 1, (2, 4): 0, (3, 4): 0} - >>> from sage.all import * >>> IncidenceStructure([[Integer(1),Integer(2),Integer(3)],[Integer(1),Integer(4)]]).degrees(Integer(2)) {(1, 2): 1, (1, 3): 1, (1, 4): 1, (2, 3): 1, (2, 4): 0, (3, 4): 0} - In a Steiner triple system, all pairs have degree 1: - sage: S13 = designs.steiner_triple_system(13) sage: all(v == 1 for v in S13.degrees(2).values()) True - >>> from sage.all import * >>> S13 = designs.steiner_triple_system(Integer(13)) >>> all(v == Integer(1) for v in S13.degrees(Integer(2)).values()) True 
 - dual(algorithm=None)[source]¶
- 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_packagesSage spkg).
 - EXAMPLES: - The dual of a projective plane is a projective plane: - sage: PP = designs.DesarguesianProjectivePlaneDesign(4) # needs sage.rings.finite_rings sage: PP.dual().is_t_design(return_parameters=True) # needs sage.modules sage.rings.finite_rings (True, (2, 21, 5, 1)) - >>> from sage.all import * >>> PP = designs.DesarguesianProjectivePlaneDesign(Integer(4)) # needs sage.rings.finite_rings >>> PP.dual().is_t_design(return_parameters=True) # needs sage.modules sage.rings.finite_rings (True, (2, 21, 5, 1)) - REFERENCE: - Leonard Soicher, Design package manual 
 
 - edge_coloring()[source]¶
- 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 - >>> from sage.all import * >>> H = Hypergraph([{Integer(1),Integer(2),Integer(3)},{Integer(2),Integer(3),Integer(4)},{Integer(3),Integer(4),Integer(5)},{Integer(4),Integer(5),Integer(6)}]); H Incidence structure with 6 points and 4 blocks >>> C = H.edge_coloring() >>> C # random [[[3, 4, 5]], [[2, 3, 4]], [[4, 5, 6], [1, 2, 3]]] >>> Set(map(Set,sum(C,[]))) == Set(map(Set,H.blocks())) True 
 - ground_set()[source]¶
- Return the ground set (i.e the list of points). - EXAMPLES: - sage: IncidenceStructure(3, [[0,1],[0,2]]).ground_set() [0, 1, 2] - >>> from sage.all import * >>> IncidenceStructure(Integer(3), [[Integer(0),Integer(1)],[Integer(0),Integer(2)]]).ground_set() [0, 1, 2] 
 - incidence_graph(labels=False)[source]¶
- Return the incidence graph of the incidence structure. - A point and a block are adjacent in this graph whenever they are incident. - INPUT: - labels– boolean; whether to return a graph whose vertices are integers, or labelled elements- labels is False– default; in this case the first vertices of the graphs are the elements of- ground_set(), and appear in the same order. Similarly, the following vertices represent the elements of- blocks(), and appear in the same order.
- labels is True, the points keep their original labels, and the blocks are- Setobjects.- Note that the labelled incidence graph can be incorrect when blocks are repeated, and on some (rare) occasions when the elements of - ground_set()mix- Set()and non-- Setobjects.
 
 - EXAMPLES: - sage: BD = 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() # needs sage.modules Bipartite graph on 14 vertices sage: A = BD.incidence_matrix() # needs sage.modules sage: Graph(block_matrix([[A*0, A], # needs sage.modules ....: [A.transpose(),A*0]])) == BD.incidence_graph() True - >>> from sage.all import * >>> BD = IncidenceStructure(Integer(7), [[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)], ... [Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(5)]]) >>> BD.incidence_graph() # needs sage.modules Bipartite graph on 14 vertices >>> A = BD.incidence_matrix() # needs sage.modules >>> Graph(block_matrix([[A*Integer(0), A], # needs sage.modules ... [A.transpose(),A*Integer(0)]])) == BD.incidence_graph() True 
 - incidence_matrix()[source]¶
- Return the incidence matrix \(A\) of the design. A is a \((v \times b)\) matrix defined by: - A[i,j] = 1if- iis in block- B_jand 0 otherwise.- EXAMPLES: - sage: BD = 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() # needs sage.modules [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 = IncidenceStructure('abc', ('ab','abc','ac','c')) sage: I.incidence_matrix() # needs sage.modules [1 1 1 0] [1 1 0 0] [0 1 1 1] - >>> from sage.all import * >>> BD = IncidenceStructure(Integer(7), [[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)], ... [Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(5)]]) >>> BD.block_sizes() [3, 3, 3, 3, 3, 3, 3] >>> BD.incidence_matrix() # needs sage.modules [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] >>> I = IncidenceStructure('abc', ('ab','abc','ac','c')) >>> I.incidence_matrix() # needs sage.modules [1 1 1 0] [1 1 0 0] [0 1 1 1] 
 - induced_substructure(points)[source]¶
- Return the substructure induced by a set of points. - The substructure induced in \(\mathcal H\) by a set \(X\subseteq V(\mathcal H)\) of points is the incidence structure \(\mathcal H_X\) defined on \(X\) whose sets are all \(S\in \mathcal H\) such that \(S\subseteq X\). - INPUT: - points– set of points
 - Note - This method goes over all sets of - selfbefore building a new- IncidenceStructure(which involves some relabelling and sorting). It probably should not be called in a performance-critical code.- EXAMPLES: - A Fano plane with one point removed: - sage: F = designs.steiner_triple_system(7) sage: F.induced_substructure([0..5]) Incidence structure with 6 points and 4 blocks - >>> from sage.all import * >>> F = designs.steiner_triple_system(Integer(7)) >>> F.induced_substructure((ellipsis_range(Integer(0),Ellipsis,Integer(5)))) Incidence structure with 6 points and 4 blocks 
 - intersection_graph(sizes=None)[source]¶
- Return the intersection graph of the incidence structure. - The vertices of this graph are the - blocks()of the incidence structure. Two of them are adjacent if the size of their intersection belongs to the set- sizes.- INPUT: - sizes– list/set of integers; for convenience, setting- sizesto- 5has the same effect as- sizes=[5]. When set to- None(default), behaves as- sizes=PositiveIntegers().
 - EXAMPLES: - The intersection graph of a - balanced_incomplete_block_design()is a- strongly regular graph(when it is not trivial):- sage: BIBD = designs.balanced_incomplete_block_design(19,3) sage: G = BIBD.intersection_graph(1) sage: G.is_strongly_regular(parameters=True) (57, 24, 11, 9) - >>> from sage.all import * >>> BIBD = designs.balanced_incomplete_block_design(Integer(19),Integer(3)) >>> G = BIBD.intersection_graph(Integer(1)) >>> G.is_strongly_regular(parameters=True) (57, 24, 11, 9) 
 - is_berge_cyclic()[source]¶
- Check whether - selfis a Berge-Cyclic uniform hypergraph.- A \(k\)-uniform Berge cycle (named after Claude Berge) of length \(\ell\) is a cyclic list of distinct \(k\)-sets \(F_1,\ldots,F_\ell\), \(\ell>1\), and distinct vertices \(C = \{v_1,\ldots,v_\ell\}\) such that for each \(1\le i\le \ell\), \(F_i\) contains \(v_i\) and \(v_{i+1}\) (where \(v_{l+1} = v_1\)). - A uniform hypergraph is Berge-cyclic if its incidence graph is cyclic. It is called “Berge-acyclic” otherwise. - For more information, see [Fag1983] and Wikipedia article Hypergraph. - EXAMPLES: - sage: Hypergraph(5, [[1, 2, 3], [2, 3, 4]]).is_berge_cyclic() # needs sage.modules True sage: Hypergraph(6, [[1, 2, 3], [3, 4, 5]]).is_berge_cyclic() # needs sage.modules False - >>> from sage.all import * >>> Hypergraph(Integer(5), [[Integer(1), Integer(2), Integer(3)], [Integer(2), Integer(3), Integer(4)]]).is_berge_cyclic() # needs sage.modules True >>> Hypergraph(Integer(6), [[Integer(1), Integer(2), Integer(3)], [Integer(3), Integer(4), Integer(5)]]).is_berge_cyclic() # needs sage.modules False 
 - is_connected()[source]¶
- Test whether the design is connected. - EXAMPLES: - sage: IncidenceStructure(3, [[0,1],[0,2]]).is_connected() True sage: IncidenceStructure(4, [[0,1],[2,3]]).is_connected() False - >>> from sage.all import * >>> IncidenceStructure(Integer(3), [[Integer(0),Integer(1)],[Integer(0),Integer(2)]]).is_connected() True >>> IncidenceStructure(Integer(4), [[Integer(0),Integer(1)],[Integer(2),Integer(3)]]).is_connected() False 
 - is_generalized_quadrangle(verbose=False, parameters=False)[source]¶
- Test if the incidence structure is a generalized quadrangle. - An incidence structure is a generalized quadrangle iff (see [BH2012], section 9.6): - two blocks intersect on at most one point. 
- For every point \(p\) not in a block \(B\), there is a unique block \(B'\) intersecting both \(\{p\}\) and \(B\) 
 - It is a regular generalized quadrangle if furthermore: - it is \(s+1\)- - uniformfor some positive integer \(s\).
- it is \(t+1\)- - regularfor some positive integer \(t\).
 - For more information, see the Wikipedia article Generalized_quadrangle. - Note - Some references (e.g. [PT2009] or Wikipedia article Generalized_quadrangle) only allow regular generalized quadrangles. To use such a definition, see the - parametersoptional argument described below, or the methods- is_regular()and- is_uniform().- INPUT: - verbose– boolean; whether to print an explanation when the instance is not a generalized quadrangle
- parameters– (boolean;- False); if set to- True, the function returns a pair- (s,t)instead of- Trueanswers. In this case, \(s\) and \(t\) are the integers defined above if they exist (each can be set to- Falseotherwise).
 - EXAMPLES: - sage: h = designs.CremonaRichmondConfiguration() # needs networkx sage: h.is_generalized_quadrangle() # needs networkx True - >>> from sage.all import * >>> h = designs.CremonaRichmondConfiguration() # needs networkx >>> h.is_generalized_quadrangle() # needs networkx True - This is actually a regular generalized quadrangle: - sage: h.is_generalized_quadrangle(parameters=True) # needs networkx (2, 2) - >>> from sage.all import * >>> h.is_generalized_quadrangle(parameters=True) # needs networkx (2, 2) 
 - is_isomorphic(other, certificate=False)[source]¶
- Return whether the two incidence structures are isomorphic. - INPUT: - other– an incidence structure
- certificate– boolean (default:- False); whether to return an isomorphism from- selfto- otherinstead of a boolean answer
 - EXAMPLES: - sage: # needs sage.schemes sage: fano1 = designs.balanced_incomplete_block_design(7,3) sage: fano2 = designs.projective_plane(2) sage: fano1.is_isomorphic(fano2) True sage: fano1.is_isomorphic(fano2,certificate=True) {0: 0, 1: 1, 2: 2, 3: 6, 4: 4, 5: 3, 6: 5} - >>> from sage.all import * >>> # needs sage.schemes >>> fano1 = designs.balanced_incomplete_block_design(Integer(7),Integer(3)) >>> fano2 = designs.projective_plane(Integer(2)) >>> fano1.is_isomorphic(fano2) True >>> fano1.is_isomorphic(fano2,certificate=True) {0: 0, 1: 1, 2: 2, 3: 6, 4: 4, 5: 3, 6: 5} 
 - is_regular(r=None)[source]¶
- Test whether the incidence structure is \(r\)-regular. - An incidence structure is said to be \(r\)-regular if all its points are incident with exactly \(r\) blocks. - INPUT: - r– integer
 - OUTPUT: - If - ris defined, a boolean is returned. If- ris set to- None(default), the method returns either- Falseor the integer- rsuch that the incidence structure is \(r\)-regular.- Warning - In case of \(0\)-regular incidence structure, beware that - if not H.is_regular()is a satisfied condition.- EXAMPLES: - sage: designs.balanced_incomplete_block_design(7,3).is_regular() # needs sage.schemes 3 sage: designs.balanced_incomplete_block_design(7,3).is_regular(r=3) # needs sage.schemes True sage: designs.balanced_incomplete_block_design(7,3).is_regular(r=4) # needs sage.schemes False - >>> from sage.all import * >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_regular() # needs sage.schemes 3 >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_regular(r=Integer(3)) # needs sage.schemes True >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_regular(r=Integer(4)) # needs sage.schemes False 
 - is_resolvable(certificate, solver=False, verbose=None, check=0, integrality_tolerance=True)[source]¶
- Test whether the hypergraph is resolvable. - A hypergraph is said to be resolvable if its sets can be partitionned into classes, each of which is a partition of the ground set. - Note - This problem is solved using an Integer Linear Program, and GLPK (the default LP solver) has been reported to be very slow on some instances. If you hit this wall, consider installing a more powerful MILP solver (CPLEX, Gurobi, …). - INPUT: - certificate– boolean; whether to return the classes along with the binary answer (see examples below)
- solver– (default:- None) specify a Mixed Integer Linear Programming (MILP) solver to be used. If set to- None, the default one is used. For more information on MILP solvers and which default solver is used, see the method- solveof the class- MixedIntegerLinearProgram.
- verbose– integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
- check– boolean (default:- True); whether to check that output is correct before returning it. As this is expected to be useless, you may want to disable it whenever you want speed.
- integrality_tolerance– parameter for use with MILP solvers over an inexact base ring; see- MixedIntegerLinearProgram.get_values()
 - EXAMPLES: - Some resolvable designs: - sage: TD = designs.transversal_design(2,2,resolvable=True) sage: TD.is_resolvable() True sage: AG = designs.AffineGeometryDesign(3,1,GF(2)) # needs sage.combinat sage: AG.is_resolvable() # needs sage.combinat True - >>> from sage.all import * >>> TD = designs.transversal_design(Integer(2),Integer(2),resolvable=True) >>> TD.is_resolvable() True >>> AG = designs.AffineGeometryDesign(Integer(3),Integer(1),GF(Integer(2))) # needs sage.combinat >>> AG.is_resolvable() # needs sage.combinat True - Their classes: - sage: b, cls = TD.is_resolvable(True) sage: b True sage: cls # random [[[0, 3], [1, 2]], [[1, 3], [0, 2]]] sage: # needs sage.combinat sage: b, cls = AG.is_resolvable(True) sage: b True sage: cls # random [[[6, 7], [4, 5], [0, 1], [2, 3]], [[5, 7], [0, 4], [3, 6], [1, 2]], [[0, 2], [4, 7], [1, 3], [5, 6]], [[3, 4], [0, 7], [1, 5], [2, 6]], [[3, 7], [1, 6], [0, 5], [2, 4]], [[0, 6], [2, 7], [1, 4], [3, 5]], [[4, 6], [0, 3], [2, 5], [1, 7]]] - >>> from sage.all import * >>> b, cls = TD.is_resolvable(True) >>> b True >>> cls # random [[[0, 3], [1, 2]], [[1, 3], [0, 2]]] >>> # needs sage.combinat >>> b, cls = AG.is_resolvable(True) >>> b True >>> cls # random [[[6, 7], [4, 5], [0, 1], [2, 3]], [[5, 7], [0, 4], [3, 6], [1, 2]], [[0, 2], [4, 7], [1, 3], [5, 6]], [[3, 4], [0, 7], [1, 5], [2, 6]], [[3, 7], [1, 6], [0, 5], [2, 4]], [[0, 6], [2, 7], [1, 4], [3, 5]], [[4, 6], [0, 3], [2, 5], [1, 7]]] - A non-resolvable design: - sage: Fano = designs.balanced_incomplete_block_design(7,3) # needs sage.schemes sage: Fano.is_resolvable() # needs sage.schemes False sage: Fano.is_resolvable(True) # needs sage.schemes (False, []) - >>> from sage.all import * >>> Fano = designs.balanced_incomplete_block_design(Integer(7),Integer(3)) # needs sage.schemes >>> Fano.is_resolvable() # needs sage.schemes False >>> Fano.is_resolvable(True) # needs sage.schemes (False, []) 
 - is_simple()[source]¶
- Test whether this design is simple (i.e. no repeated block). - EXAMPLES: - sage: IncidenceStructure(3, [[0,1],[1,2],[0,2]]).is_simple() True sage: 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 = IncidenceStructure(V, B) sage: I.is_simple() True sage: I2 = IncidenceStructure(V, B*2) sage: I2.is_simple() False - >>> from sage.all import * >>> IncidenceStructure(Integer(3), [[Integer(0),Integer(1)],[Integer(1),Integer(2)],[Integer(0),Integer(2)]]).is_simple() True >>> IncidenceStructure(Integer(3), [[Integer(0)],[Integer(0)]]).is_simple() False >>> V = [(Integer(0),'a'),(Integer(0),'b'),(Integer(1),'a'),(Integer(1),'b')] >>> B = [[V[Integer(0)],V[Integer(1)]], [V[Integer(1)],V[Integer(2)]]] >>> I = IncidenceStructure(V, B) >>> I.is_simple() True >>> I2 = IncidenceStructure(V, B*Integer(2)) >>> I2.is_simple() False 
 - is_spread(spread)[source]¶
- Check whether the input is a spread for - self.- A spread of an incidence structure \((P, B)\) is a subset of \(B\) which forms a partition of \(P\). - INPUT: - spread– iterable; defines the spread
 - EXAMPLES: - sage: E = IncidenceStructure([[1, 2, 3], [4, 5, 6], [1, 5, 6]]) sage: E.is_spread([[1, 2, 3], [4, 5, 6]]) True sage: E.is_spread([1, 2, 3, 4, 5, 6]) Traceback (most recent call last): ... TypeError: 'sage.rings.integer.Integer' object is not iterable sage: E.is_spread([[1, 2, 3, 4], [5, 6]]) False - >>> from sage.all import * >>> E = IncidenceStructure([[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(1), Integer(5), Integer(6)]]) >>> E.is_spread([[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)]]) True >>> E.is_spread([Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)]) Traceback (most recent call last): ... TypeError: 'sage.rings.integer.Integer' object is not iterable >>> E.is_spread([[Integer(1), Integer(2), Integer(3), Integer(4)], [Integer(5), Integer(6)]]) False - Order of blocks or of points within each block doesn’t matter: - sage: E = IncidenceStructure([[1, 2, 3], [4, 5, 6], [1, 5, 6]]) sage: E.is_spread([[5, 6, 4], [3, 1, 2]]) True - >>> from sage.all import * >>> E = IncidenceStructure([[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(1), Integer(5), Integer(6)]]) >>> E.is_spread([[Integer(5), Integer(6), Integer(4)], [Integer(3), Integer(1), Integer(2)]]) True 
 - is_t_design(t=None, v=None, k=None, l=None, return_parameters=False)[source]¶
- Test whether - selfis 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- Noneby default. The function tests whether the design is a \(t-(v,k,l)\) design using the provided values and guesses the others. Note that- lcannot be specified if- tis 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 = 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)) # needs sage.combinat sage: BD.is_t_design(1) # needs sage.combinat True sage: BD.is_t_design(2) # needs sage.combinat True - >>> from sage.all import * >>> fano_blocks = [[Integer(0),Integer(1),Integer(2)],[Integer(0),Integer(3),Integer(4)],[Integer(0),Integer(5),Integer(6)],[Integer(1),Integer(3),Integer(5)],[Integer(1),Integer(4),Integer(6)],[Integer(2),Integer(3),Integer(6)],[Integer(2),Integer(4),Integer(5)]] >>> BD = IncidenceStructure(Integer(7), fano_blocks) >>> BD.is_t_design() True >>> BD.is_t_design(return_parameters=True) (True, (2, 7, 3, 1)) >>> BD.is_t_design(Integer(2), Integer(7), Integer(3), Integer(1)) True >>> BD.is_t_design(Integer(1), Integer(7), Integer(3), Integer(3)) True >>> BD.is_t_design(Integer(0), Integer(7), Integer(3), Integer(7)) True >>> BD.is_t_design(Integer(0),Integer(6),Integer(3),Integer(7)) or BD.is_t_design(Integer(0),Integer(7),Integer(4),Integer(7)) or BD.is_t_design(Integer(0),Integer(7),Integer(3),Integer(8)) False >>> BD = designs.AffineGeometryDesign(Integer(3), Integer(1), GF(Integer(2))) # needs sage.combinat >>> BD.is_t_design(Integer(1)) # needs sage.combinat True >>> BD.is_t_design(Integer(2)) # needs sage.combinat 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 = IncidenceStructure(8, blocks) sage: S4_8.is_t_design(3,8,4,1) True sage: blocks = designs.steiner_quadruple_system(14) sage: S4_14 = IncidenceStructure(14, blocks) sage: S4_14.is_t_design(3,14,4,1) True - >>> from sage.all import * >>> S3_9 = designs.steiner_triple_system(Integer(9)) >>> S3_9.is_t_design(Integer(2),Integer(9),Integer(3),Integer(1)) True >>> blocks = designs.steiner_quadruple_system(Integer(8)) >>> S4_8 = IncidenceStructure(Integer(8), blocks) >>> S4_8.is_t_design(Integer(3),Integer(8),Integer(4),Integer(1)) True >>> blocks = designs.steiner_quadruple_system(Integer(14)) >>> S4_14 = IncidenceStructure(Integer(14), blocks) >>> S4_14.is_t_design(Integer(3),Integer(14),Integer(4),Integer(1)) True - Some examples of Witt designs that need the gap database: - sage: # optional - gap_package_design sage: BD = designs.WittDesign(9) sage: BD.is_t_design(2,9,3,1) True sage: W12 = designs.WittDesign(12) sage: W12.is_t_design(5,12,6,1) True sage: W12.is_t_design(4) True - >>> from sage.all import * >>> # optional - gap_package_design >>> BD = designs.WittDesign(Integer(9)) >>> BD.is_t_design(Integer(2),Integer(9),Integer(3),Integer(1)) True >>> W12 = designs.WittDesign(Integer(12)) >>> W12.is_t_design(Integer(5),Integer(12),Integer(6),Integer(1)) True >>> W12.is_t_design(Integer(4)) True - Further examples: - sage: D = IncidenceStructure(4,[[],[]]) sage: D.is_t_design(return_parameters=True) (True, (0, 4, 0, 2)) sage: D = IncidenceStructure(4, [[0,1],[0,2],[0,3]]) sage: D.is_t_design(return_parameters=True) (True, (0, 4, 2, 3)) sage: D = IncidenceStructure(4, [[0],[1],[2],[3]]) sage: D.is_t_design(return_parameters=True) (True, (1, 4, 1, 1)) sage: D = IncidenceStructure(4,[[0,1],[2,3]]) sage: D.is_t_design(return_parameters=True) (True, (1, 4, 2, 1)) sage: D = IncidenceStructure(4, [list(range(4))]) sage: D.is_t_design(return_parameters=True) (True, (4, 4, 4, 1)) - >>> from sage.all import * >>> D = IncidenceStructure(Integer(4),[[],[]]) >>> D.is_t_design(return_parameters=True) (True, (0, 4, 0, 2)) >>> D = IncidenceStructure(Integer(4), [[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(0),Integer(3)]]) >>> D.is_t_design(return_parameters=True) (True, (0, 4, 2, 3)) >>> D = IncidenceStructure(Integer(4), [[Integer(0)],[Integer(1)],[Integer(2)],[Integer(3)]]) >>> D.is_t_design(return_parameters=True) (True, (1, 4, 1, 1)) >>> D = IncidenceStructure(Integer(4),[[Integer(0),Integer(1)],[Integer(2),Integer(3)]]) >>> D.is_t_design(return_parameters=True) (True, (1, 4, 2, 1)) >>> D = IncidenceStructure(Integer(4), [list(range(Integer(4)))]) >>> D.is_t_design(return_parameters=True) (True, (4, 4, 4, 1)) 
 - is_uniform(k=None)[source]¶
- Test whether the incidence structure is \(k\)-uniform - An incidence structure is said to be \(k\)-uniform if all its blocks have size \(k\). - INPUT: - k– integer
 - OUTPUT: - If - kis defined, a boolean is returned. If- kis set to- None(default), the method returns either- Falseor the integer- ksuch that the incidence structure is \(k\)-uniform.- Warning - In case of \(0\)-uniform incidence structure, beware that - if not H.is_uniform()is a satisfied condition.- EXAMPLES: - sage: designs.balanced_incomplete_block_design(7,3).is_uniform() # needs sage.schemes 3 sage: designs.balanced_incomplete_block_design(7,3).is_uniform(k=3) # needs sage.schemes True sage: designs.balanced_incomplete_block_design(7,3).is_uniform(k=4) # needs sage.schemes False - >>> from sage.all import * >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_uniform() # needs sage.schemes 3 >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_uniform(k=Integer(3)) # needs sage.schemes True >>> designs.balanced_incomplete_block_design(Integer(7),Integer(3)).is_uniform(k=Integer(4)) # needs sage.schemes False 
 - isomorphic_substructures_iterator(H2, induced=False)[source]¶
- Iterate over all copies of - H2contained in- self.- A hypergraph \(H_1\) contains an isomorphic copy of a hypergraph \(H_2\) if there exists an injection \(f:V(H_2)\mapsto V(H_1)\) such that for any set \(S_2\in E(H_2)\) the set \(S_1=f(S2)\) belongs to \(E(H_1)\). - It is an induced copy if no other set of \(E(H_1)\) is contained in \(f(V(H_2))\), i.e. \(|E(H_2)|=\{S:S\in E(H_1)\text{ and }f(V(H_2))\}\). - This function lists all such injections. In particular, the number of copies of \(H\) in itself is equal to the size of its automorphism group. - See - subhypergraph_searchfor more information.- INPUT: - H2– an- IncidenceStructureobject
- induced– boolean (default:- False); whether to require the copies to be induced
 - EXAMPLES: - How many distinct \(C_5\) in Petersen’s graph ? - sage: P = graphs.PetersenGraph() sage: C = graphs.CycleGraph(5) sage: IP = IncidenceStructure(P.edges(sort=True, labels=False)) sage: IC = IncidenceStructure(C.edges(sort=True, labels=False)) sage: sum(1 for _ in IP.isomorphic_substructures_iterator(IC)) 120 - >>> from sage.all import * >>> P = graphs.PetersenGraph() >>> C = graphs.CycleGraph(Integer(5)) >>> IP = IncidenceStructure(P.edges(sort=True, labels=False)) >>> IC = IncidenceStructure(C.edges(sort=True, labels=False)) >>> sum(Integer(1) for _ in IP.isomorphic_substructures_iterator(IC)) 120 - As the automorphism group of \(C_5\) has size 10, the number of distinct unlabelled copies is 12. Let us check that all functions returned correspond to an actual \(C_5\) subgraph: - sage: for f in IP.isomorphic_substructures_iterator(IC): ....: assert all(P.has_edge(f[x],f[y]) for x,y in C.edges(sort=True, labels=False)) - >>> from sage.all import * >>> for f in IP.isomorphic_substructures_iterator(IC): ... assert all(P.has_edge(f[x],f[y]) for x,y in C.edges(sort=True, labels=False)) - The number of induced copies, in this case, is the same: - sage: sum(1 for _ in IP.isomorphic_substructures_iterator(IC,induced=True)) 120 - >>> from sage.all import * >>> sum(Integer(1) for _ in IP.isomorphic_substructures_iterator(IC,induced=True)) 120 - They begin to differ if we make one vertex universal: - sage: P.add_edges([(0,x) for x in P], loops=False) sage: IP = IncidenceStructure(P.edges(sort=True, labels=False)) sage: IC = IncidenceStructure(C.edges(sort=True, labels=False)) sage: sum(1 for _ in IP.isomorphic_substructures_iterator(IC)) 420 sage: sum(1 for _ in IP.isomorphic_substructures_iterator(IC,induced=True)) 60 - >>> from sage.all import * >>> P.add_edges([(Integer(0),x) for x in P], loops=False) >>> IP = IncidenceStructure(P.edges(sort=True, labels=False)) >>> IC = IncidenceStructure(C.edges(sort=True, labels=False)) >>> sum(Integer(1) for _ in IP.isomorphic_substructures_iterator(IC)) 420 >>> sum(Integer(1) for _ in IP.isomorphic_substructures_iterator(IC,induced=True)) 60 - The number of copies of \(H\) in itself is the size of its automorphism group: - sage: H = designs.projective_plane(3) # needs sage.schemes sage: sum(1 for _ in H.isomorphic_substructures_iterator(H)) # needs sage.schemes 5616 sage: H.automorphism_group().cardinality() # needs sage.groups sage.schemes 5616 - >>> from sage.all import * >>> H = designs.projective_plane(Integer(3)) # needs sage.schemes >>> sum(Integer(1) for _ in H.isomorphic_substructures_iterator(H)) # needs sage.schemes 5616 >>> H.automorphism_group().cardinality() # needs sage.groups sage.schemes 5616 
 - num_blocks()[source]¶
- Return the number of blocks. - EXAMPLES: - sage: designs.DesarguesianProjectivePlaneDesign(2).num_blocks() 7 sage: B = IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]]) sage: B.num_blocks() 5 - >>> from sage.all import * >>> designs.DesarguesianProjectivePlaneDesign(Integer(2)).num_blocks() 7 >>> B = IncidenceStructure(Integer(4), [[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(0),Integer(3)],[Integer(1),Integer(2)], [Integer(1),Integer(2),Integer(3)]]) >>> B.num_blocks() 5 
 - num_points()[source]¶
- Return the size of the ground set. - EXAMPLES: - sage: designs.DesarguesianProjectivePlaneDesign(2).num_points() 7 sage: B = IncidenceStructure(4, [[0,1],[0,2],[0,3],[1,2], [1,2,3]]) sage: B.num_points() 4 - >>> from sage.all import * >>> designs.DesarguesianProjectivePlaneDesign(Integer(2)).num_points() 7 >>> B = IncidenceStructure(Integer(4), [[Integer(0),Integer(1)],[Integer(0),Integer(2)],[Integer(0),Integer(3)],[Integer(1),Integer(2)], [Integer(1),Integer(2),Integer(3)]]) >>> B.num_points() 4 
 - packing(solver, verbose=None, integrality_tolerance=0)[source]¶
- 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 Mixed Integer Linear Programming (MILP) 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- solveof the class- MixedIntegerLinearProgram.
- verbose– integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.
- integrality_tolerance– parameter for use with MILP solvers over an inexact base ring; see- MixedIntegerLinearProgram.get_values().
 - EXAMPLES: - sage: P = IncidenceStructure([[1,2],[3,4],[2,3]]).packing() # needs sage.numerical.mip sage: sorted(sorted(b) for b in P) # needs sage.numerical.mip [[1, 2], [3, 4]] sage: len(designs.steiner_triple_system(9).packing()) # needs sage.numerical.mip 3 - >>> from sage.all import * >>> P = IncidenceStructure([[Integer(1),Integer(2)],[Integer(3),Integer(4)],[Integer(2),Integer(3)]]).packing() # needs sage.numerical.mip >>> sorted(sorted(b) for b in P) # needs sage.numerical.mip [[1, 2], [3, 4]] >>> len(designs.steiner_triple_system(Integer(9)).packing()) # needs sage.numerical.mip 3 
 - rank()[source]¶
- Return the rank of the hypergraph (the maximum size of a block). - EXAMPLES: - sage: h = Hypergraph(8, [[0,1,3],[1,4,5,6],[1,2]]) sage: h.rank() 4 - >>> from sage.all import * >>> h = Hypergraph(Integer(8), [[Integer(0),Integer(1),Integer(3)],[Integer(1),Integer(4),Integer(5),Integer(6)],[Integer(1),Integer(2)]]) >>> h.rank() 4 
 - relabel(perm=None, inplace=True)[source]¶
- Relabel the ground set. - INPUT: - perm– can be one of- a dictionary – then each point - p(which should be a key of- d) is relabeled to- d[p]
- a list or a tuple of length - n– the first point returned by- ground_set()is relabeled to- l[0], the second to- l[1], …
- None– the incidence structure is relabeled to be on \(\{0,1,...,n-1\}\) in the ordering given by- ground_set()
 
- inplace– boolean (default:- False); if- Truethen return a relabeled graph and does not touch- self
 - EXAMPLES: - sage: # needs sage.schemes sage: TD = designs.transversal_design(5,5) sage: TD.relabel({i: chr(97+i) for i in range(25)}) sage: TD.ground_set() ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y'] sage: TD.blocks()[:3] [['a', 'f', 'k', 'p', 'u'], ['a', 'g', 'm', 's', 'y'], ['a', 'h', 'o', 'q', 'x']] - >>> from sage.all import * >>> # needs sage.schemes >>> TD = designs.transversal_design(Integer(5),Integer(5)) >>> TD.relabel({i: chr(Integer(97)+i) for i in range(Integer(25))}) >>> TD.ground_set() ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y'] >>> TD.blocks()[:Integer(3)] [['a', 'f', 'k', 'p', 'u'], ['a', 'g', 'm', 's', 'y'], ['a', 'h', 'o', 'q', 'x']] - Relabel to integer points: - sage: TD.relabel() # needs sage.schemes sage: TD.blocks()[:3] # needs sage.schemes [[0, 5, 10, 15, 20], [0, 6, 12, 18, 24], [0, 7, 14, 16, 23]] - >>> from sage.all import * >>> TD.relabel() # needs sage.schemes >>> TD.blocks()[:Integer(3)] # needs sage.schemes [[0, 5, 10, 15, 20], [0, 6, 12, 18, 24], [0, 7, 14, 16, 23]] 
 - trace(points, min_size=1, multiset=True)[source]¶
- Return the trace of a set of points. - Given an hypergraph \(\mathcal H\), the trace of a set \(X\) of points in \(\mathcal H\) is the hypergraph whose blocks are all non-empty \(S \cap X\) where \(S \in \mathcal H\). - INPUT: - points– set of points
- min_size– integer (default: 1); minimum size of the sets to keep. By default all empty sets are discarded, i.e.- min_size=1
- multiset– boolean (default:- True); whether to keep multiple copies of the same set
 - Note - This method goes over all sets of - selfbefore building a new- IncidenceStructure(which involves some relabelling and sorting). It probably should not be called in a performance-critical code.- EXAMPLES: - A Baer subplane of order 2 (i.e. a Fano plane) in a projective plane of order 4: - sage: # needs sage.schemes sage: P4 = designs.projective_plane(4) sage: F = designs.projective_plane(2) sage: for x in Subsets(P4.ground_set(),7): ....: if P4.trace(x,min_size=2).is_isomorphic(F): ....: break sage: subplane = P4.trace(x,min_size=2); subplane Incidence structure with 7 points and 7 blocks sage: subplane.is_isomorphic(F) True - >>> from sage.all import * >>> # needs sage.schemes >>> P4 = designs.projective_plane(Integer(4)) >>> F = designs.projective_plane(Integer(2)) >>> for x in Subsets(P4.ground_set(),Integer(7)): ... if P4.trace(x,min_size=Integer(2)).is_isomorphic(F): ... break >>> subplane = P4.trace(x,min_size=Integer(2)); subplane Incidence structure with 7 points and 7 blocks >>> subplane.is_isomorphic(F) True