isLinearSubclass -- whether a list of hyperplanes of a matroid is a linear subclass

Synopsis

• Usage:

  isLinearSubclass(M, H)

• Inputs:
  • M, a matroid
  • H, a list, a list of hyperplanes of the matroid
• Outputs:
  • a Boolean value, whether the list of hyperplanes is a linear subclass

Description

This function is provided by the package Matroids.

A linear subclass for a matroid M is a collection of hyperplanes H of M that uniquely determines a given modular cut of M. Specifically, H is a linear subclass if for any hyperplanes h and h' in H whose intersection has rank equal to the rank of M minus two, every hyperplane of M containing the intersection of h and h' is also in H.

Each hyperplane in the linear subclass may be represented as either a set or a list of elements of the ground set of M. The list of hyperplanes H below is a linear subclass for the uniform matroid U_{4,6} since no two hyperplanes in H have an intersection of rank 2.

i1 : M = uniformMatroid(4, 6);

i2 : H = {set{0,1,2}, set{2,3,4}, set{0,3,5}};

i3 : isLinearSubclass(M, H)

o3 = true

The list of hyperplanes H' below is not a linear subclass since the hyperplanes {0,1,2} and {0,1,4} have an intersection of rank 2 which is contained in the hyperplane {0,1,3} that does not belong to H'.

i4 : H' = {{0,1,2}, {2,3,4}, {0,3,5}, {0,1,4}};

i5 : isLinearSubclass(M, H')

o5 = false

See also

• modularCut -- associated to an elementary quotient or linear subclass
• linearSubclass -- associated to an elementary quotient or modular cut