matroid(...,EntryMode=>...) -- select method of specifying matroid

Synopsis

Usage:

matroid(..., EntryMode => "bases")

matroid(..., EntryMode => "nonbases")

matroid(..., EntryMode => "circuits")

Description

A matroid is determined by its set of bases, i.e. maximal (with respect to inclusion) independent sets, which are all of the same size (namely, the rank of the matroid). However, many interesting matroids have relatively few dependencies, and thus it may be easier to specify the matroid by its nonbases, i.e. dependent subsets of the ground set, with size equal to the rank of the matroid.

Similarly, a matroid can be specified by its circuits, i.e. minimal dependent sets. This is done e.g. when creating a graphical matroid.

If EntryMode is not specified, then the default value is assumed, which is EntryMode => "bases".

i1 : M = matroid({{0,1,2}, {3,4,5}}, EntryMode => "circuits") -- bowtie graph / 2 disjoint K3's

o1 = a "matroid" of rank 4 on 6 elements

o1 : Matroid

i2 : bases M

o2 = {set {1, 2, 4, 5}, set {0, 2, 4, 5}, set {0, 1, 4, 5}, set {1, 2, 3, 5},
     ------------------------------------------------------------------------
     set {0, 2, 3, 5}, set {0, 1, 3, 5}, set {1, 2, 3, 4}, set {0, 2, 3, 4},
     ------------------------------------------------------------------------
     set {0, 1, 3, 4}}

o2 : List

i3 : F7 = matroid({{0,1,6},{0,2,4},{0,3,5},{1,2,5},{1,3,4},{2,3,6},{4,5,6}}, EntryMode => "nonbases")

o3 = a "matroid" of rank 3 on 7 elements

o3 : Matroid

i4 : F7 == specificMatroid "fano"

o4 = true

Further information

Default value: bases
Function: matroid -- constructs a matroid
Option key: EntryMode -- Constructs a simple graph

Functions with optional argument named EntryMode :

digraph(...,EntryMode=>...) (missing documentation)
elementaryQuotient(...,EntryMode=>...) -- use a modular cut or linear subclass
extension(...,EntryMode=>...) -- use a modular cut or linear subclass
graph(...,EntryMode=>...) -- see graph -- Constructs a simple graph
matroid(...,EntryMode=>...) -- select method of specifying matroid

matroid(...,EntryMode=>...) -- select method of specifying matroid

Synopsis

Description

Further information

See also

Functions with optional argument named EntryMode :