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Matroids :: matroid(...,EntryMode=>...)

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

Synopsis

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

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

o3 = a matroid of rank 3 on 7 elements

o3 : Matroid
i4 : F7 == specificMatroid "fano"

o4 = false

Further information

See also

Functions with optional argument named EntryMode :