CurrentModule = Oscar
Matroids are a fundamental combinatorial object with connections to various fields of mathematics. It is an abstraction of linear independence in vector spaces and forests in graphs. One way to define a matroid is via the following two sets of data:
- a finite ground set
$E := {1,\ldots,n}$ and - a nonempty finite set
$\mathcal{B} \subseteq \mathcal{P}(E)$ of bases satisfying an exchange property.
There are however many equivalent ways to define a matroid. One can also define a matroid via its circuits, hyperplanes, a graph, or a matrix. For a detailed introduction of matroids we refer to the textbook Oxl11.
matroid_from_bases(bases::AbstractVector{T}, nelements::IntegerUnion; check::Bool=true) where T<:GroundsetType
matroid_from_nonbases(nonbases::AbstractVector{T}, nelements::IntegerUnion; check::Bool=true) where T<:GroundsetType
matroid_from_circuits(circuits::AbstractVector{T}, nelements::IntegerUnion) where T<:GroundsetType
matroid_from_hyperplanes(hyperplanes::AbstractVector{T}, nelements::IntegerUnion) where T<:GroundsetType
matroid_from_matrix_columns(A::MatrixElem; check::Bool=true)
matroid_from_matrix_rows(A::MatrixElem, ; check::Bool=true)
cycle_matroid
bond_matroid(g::Graph)
cocycle_matroid(g::Graph)
Matroid(pm_matroid::Polymake.BigObjectAllocated, E::GroundsetType=Vector{Integer}(1:pm_matroid.N_ELEMENTS))
matroid_from_revlex_basis_encoding(rvlx::String, r::IntegerUnion, n::IntegerUnion)
uniform_matroid(r::IntegerUnion,n::IntegerUnion)
fano_matroid()
moebius_kantor_matroid()
non_fano_matroid()
non_pappus_matroid()
pappus_matroid()
perles_matroid()
R10_matroid()
vamos_matroid()
all_subsets_matroid(r::Int)
projective_plane(q::Int)
projective_geometry(r::Int, q::Int; check::Bool=false)
affine_geometry(r::Int, q::Int; check::Bool=false)
dual_matroid(M::Matroid)
direct_sum(M::Matroid, N::Matroid)
deletion(M::Matroid,set::GroundsetType)
restriction(M::Matroid, set::GroundsetType)
contraction(M::Matroid,set::GroundsetType)
minor(M::Matroid, set_del::GroundsetType, set_cont::GroundsetType)
principal_extension(M::Matroid, set::GroundsetType, elem::ElementType)
free_extension(M::Matroid, elem::ElementType)
series_extension(M::Matroid, old::ElementType, new::ElementType)
parallel_extension(M::Matroid, old::ElementType, new::ElementType)
matroid_groundset(M::Matroid)
length(M::Matroid)
rank(M::Matroid)
rank(M::Matroid, set::GroundsetType)
bases(M::Matroid)
nonbases(M::Matroid)
circuits(M::Matroid)
hyperplanes(M::Matroid)
flats(M::Matroid, r::Union{Int,Nothing}=nothing)
cyclic_flats(M::Matroid, r::Union{Int,Nothing}=nothing)
closure(M::Matroid, set::GroundsetType)
nullity(M::Matroid, set::GroundsetType)
fundamental_circuit(M::Matroid, basis::GroundsetType, elem::ElementType)
fundamental_cocircuit(M::Matroid, basis::GroundsetType, elem::ElementType)
independent_sets(M::Matroid)
spanning_sets(M::Matroid)
cobases(M::Matroid)
cocircuits(M::Matroid)
cohyperplanes(M::Matroid)
corank(M::Matroid, set::GroundsetType)
is_clutter(sets::AbstractVector{T}) where T <: GroundsetType
is_regular(M::Matroid)
is_binary(M::Matroid)
is_ternary(M::Matroid)
n_connected_components(M::Matroid)
connected_components(M::Matroid)
is_connected(M::Matroid)
loops(M::Matroid)
coloops(M::Matroid)
is_loopless(M::Matroid)
is_coloopless(M::Matroid)
is_simple(M::Matroid)
direct_sum_components(M::Matroid)
connectivity_function(M::Matroid, set::GroundsetType)
is_vertical_k_separation(M::Matroid,k::IntegerUnion, set::GroundsetType)
is_k_separation(M::Matroid,k::IntegerUnion, set::GroundsetType)
vertical_connectivity(M::Matroid)
girth(M::Matroid, set::GroundsetType=M.groundset)
tutte_connectivity(M::Matroid)
tutte_polynomial(M::Matroid)
characteristic_polynomial(M::Matroid)
reduced_characteristic_polynomial(M::Matroid)
revlex_basis_encoding(M::Matroid)
is_isomorphic(M1::Matroid, M2::Matroid)
is_minor(M::Matroid, N::Matroid)
automorphism_group(M::Matroid)
matroid_base_polytope(M::Matroid)
chow_ring(M::Matroid; ring::Union{MPolyRing,Nothing}=nothing, extended::Bool=false)
augmented_chow_ring(M::Matroid)