next | previous | forward | backward | up | index | toc

# relaxation -- relaxation of matroid

## Synopsis

• Usage:
relaxation(M, S)
relaxation M
• Inputs:
• M, ,
• S, a set, of indices, or a list of elements in M, which is a circuit-hyperplane of M
• Optional inputs:
• Outputs:
• , the relaxation of M by S

## Description

Let M = (E, B) be a matroid with bases B. If there is a subset S of E that is both a circuit and a hyperplane of M, then the set $B \cup \{S\}$ is the set of bases of a matroid on E, called the relaxation of M by S.

If no set S is provided, then this function will take S to be a random circuit-hyperplane (the first in lexicographic order). If no circuit-hyperplanes exist, then an error is produced.

Many interesting matroids arise as relaxations of other matroids: e.g. the non-Fano matroid is a relaxation of the Fano matroid, and the non-Pappus matroid is a relaxation of the Pappus matroid.

 i1 : P = specificMatroid "pappus" o1 = a "matroid" of rank 3 on 9 elements o1 : Matroid i2 : NP = specificMatroid "nonpappus" o2 = a "matroid" of rank 3 on 9 elements o2 : Matroid i3 : NP == relaxation(P, set{6,7,8}) o3 = true

## Ways to use relaxation :

• relaxation(Matroid)
• relaxation(Matroid,List)
• relaxation(Matroid,Set)

## For the programmer

The object relaxation is .