# sum2 -- 2-sum of matroids

## Synopsis

• Usage:
sum2(M, N)
• Inputs:
• M, ,
• N, ,
• Outputs:
• , the 2-sum of M and N with basepoint 0

## Description

This function returns the 2-sum of two given matroids M and N (cf. Oxley, Section 7.1).

It is always assumed that the common basepoint of M and N is the first element in the respective ground sets, i.e. the element with index 0. (To form a 2-sum using a different basepoint, one can first relabel M and/or N.) Moreover, it is necessary that the basepoint 0 is not a loop or coloop in either M or N. Under these assumptions, the 2-sum of M and N is equal to the contraction of the seriesConnection of M and N by 0 (or alternatively, the deletion of the parallelConnection of M and N by 0).

The operation of 2-sum is important in higher matroid connectivity: a connected matroid is 3-connected iff it cannot be expressed as a 2-sum of smaller matroids.

 i1 : M = sum2(specificMatroid "fano", uniformMatroid(2,4)) o1 = a "matroid" of rank 4 on 9 elements o1 : Matroid i2 : isConnected M o2 = true i3 : is3Connected M o3 = false