# isomorphism(Matroid,Matroid) -- computes an isomorphism between isomorphic matroids

## Synopsis

• Function: isomorphism
• Usage:
isomorphism(M, N)
• Inputs:
• M, ,
• N, ,
• Outputs:
• , an isomorphism between M and N

## Description

This method computes a single isomorphism between M and N, if one exists, and returns null if no such isomorphism exists.

The output is a HashTable, where the keys are elements of the groundSet of M, and their corresponding values are elements of (the ground set of) N.

To obtain all isomorphisms between two matroids, use getIsos.

 i1 : M = matroid({a,b,c},{{a,b},{a,c}}) o1 = a matroid of rank 2 on 3 elements o1 : Matroid i2 : isomorphism(M, uniformMatroid(2,3)) -- not isomorphic i3 : (M5, M6) = (5,6)/completeGraph/matroid o3 = (a matroid of rank 4 on 10 elements, a matroid of rank 5 on 15 elements) o3 : Sequence i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7}) o4 = a matroid of rank 4 on 10 elements o4 : Matroid i5 : time isomorphism(M5, minorM6) -- used 0.0323637 seconds o5 = HashTable{0 => 1} 1 => 0 2 => 3 3 => 2 4 => 6 5 => 5 6 => 4 7 => 9 8 => 8 9 => 7 o5 : HashTable i6 : isomorphism(M5, M5) o6 = HashTable{0 => 0} 1 => 1 2 => 2 3 => 3 4 => 4 5 => 5 6 => 6 7 => 7 8 => 8 9 => 9 o6 : HashTable i7 : N = relabel M6 o7 = a matroid of rank 5 on 15 elements o7 : Matroid i8 : time phi = isomorphism(N,M6) -- used 0.946535 seconds o8 = HashTable{0 => 8 } 1 => 1 2 => 5 3 => 2 4 => 6 5 => 4 6 => 0 7 => 12 8 => 3 9 => 10 10 => 7 11 => 11 12 => 9 13 => 13 14 => 14 o8 : HashTable