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# parallelConnection -- parallel connection of two matroids

- Usage:
    `parallelConnection(M, N)`
- Inputs:
  - M, a matroid,
  - N, a matroid,
- Outputs:
  - a matroid, the parallel connection of M and N with basepoint 0

## Description

This function returns the parallel connection of two given matroids M and N (cf. Oxley, Section 7.1). Parallel connection is dual to seriesConnection: namely, the parallel connection of M and N is the dual of the series connection of M* and N*.

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 parallel connection using a different basepoint, one can first relabel M and/or N.)

    i1 : G = graph({{0,1},{1,2},{2,3},{3,4},{4,5},{5,6},{6,0},{0,2},{0,3},{0,4},{1,3},{3,5},{3,6}})

    o1 = Graph{0 => {1, 2, 3, 4, 6}   }
               1 => {0, 2, 3}
               2 => {0, 1, 3}
               3 => {0, 1, 2, 4, 5, 6}
               4 => {0, 3, 5}
               5 => {3, 4, 6}
               6 => {0, 3, 5}

    o1 : Graph

    i2 : M = matroid G

    o2 = a "matroid" of rank 6 on 13 elements

    o2 : Matroid

    i3 : parallelConnection(M, uniformMatroid(1,2))

    o3 = a "matroid" of rank 6 on 14 elements

    o3 : Matroid

## See also

- seriesConnection -- series connection of two matroids
- sum2 -- 2-sum of matroids

## Ways to use parallelConnection:

- `parallelConnection(Matroid,Matroid)`

## For the programmer

The object parallelConnection is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Matroids/doc-Matroids.m2:1973:0.