Deletion - Maple Help

Matroids

 Deletion
 construct the matroid obtained by deletion

 Calling Sequence Deletion(M,S)

Parameters

 M - S -

Description

 • Given a matroid $M$ and a subset $S$ of its ground set $E$, the deletion $M\setminus S$ of $S$ from $M$ is a matroid whose ground set is $E\setminus S$. The independent sets of $M\setminus S$ may be obtained by removing all elements of $S$ from every independent set of $M$.

Examples

 > $\mathrm{with}\left(\mathrm{Matroids}\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,1,1,0,0,0\right],\left[0,0,0,1,1,1\right],\left[1,0,0,1,0,0\right],\left[0,1,0,0,1,0\right],\left[0,0,1,0,0,2\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]$ (1)
 > $M≔\mathrm{Matroid}\left(A\right)$
 ${M}{≔}⟨\begin{array}{c}{thⅇ linⅇar matroiⅆ whosⅇ grounⅆ sⅇt is thⅇ sⅇt of column vⅇctors of thⅇ matrix:}\\ \left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]\end{array}⟩$ (2)
 > $\mathrm{IndependentSets}\left(M\right)$
 $\left[{\varnothing }{,}\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{3}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}{,}\left\{{6}\right\}{,}\left\{{1}{,}{2}\right\}{,}\left\{{1}{,}{3}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{3}{,}{4}\right\}{,}\left\{{1}{,}{5}\right\}{,}\left\{{2}{,}{5}\right\}{,}\left\{{3}{,}{5}\right\}{,}\left\{{4}{,}{5}\right\}{,}\left\{{1}{,}{6}\right\}{,}\left\{{2}{,}{6}\right\}{,}\left\{{3}{,}{6}\right\}{,}\left\{{4}{,}{6}\right\}{,}\left\{{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}\right\}{,}\left\{{1}{,}{2}{,}{4}\right\}{,}\left\{{1}{,}{3}{,}{4}\right\}{,}\left\{{2}{,}{3}{,}{4}\right\}{,}\left\{{1}{,}{2}{,}{5}\right\}{,}\left\{{1}{,}{3}{,}{5}\right\}{,}\left\{{2}{,}{3}{,}{5}\right\}{,}\left\{{1}{,}{4}{,}{5}\right\}{,}\left\{{2}{,}{4}{,}{5}\right\}{,}\left\{{3}{,}{4}{,}{5}\right\}{,}\left\{{1}{,}{2}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{6}\right\}{,}\left\{{1}{,}{4}{,}{6}\right\}{,}\left\{{2}{,}{4}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{5}{,}{6}\right\}{,}\left\{{3}{,}{5}{,}{6}\right\}{,}\left\{{4}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{4}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{5}\right\}{,}\left\{{1}{,}{3}{,}{4}{,}{5}\right\}{,}\left\{{2}{,}{3}{,}{4}{,}{5}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{4}{,}{5}{,}{6}\right\}\right]$ (3)
 > $\mathrm{M2}≔\mathrm{Deletion}\left(M,\left\{1,2\right\}\right)$
 ${\mathrm{M2}}{≔}⟨{a matroiⅆ on 4 ⅇlⅇmⅇnts with 0 circuits}⟩$ (4)
 > $\mathrm{IndependentSets}\left(\mathrm{M2}\right)$
 $\left[{\varnothing }{,}\left\{{3}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}{,}\left\{{6}\right\}{,}\left\{{3}{,}{4}\right\}{,}\left\{{3}{,}{5}\right\}{,}\left\{{4}{,}{5}\right\}{,}\left\{{3}{,}{6}\right\}{,}\left\{{4}{,}{6}\right\}{,}\left\{{5}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{5}\right\}{,}\left\{{3}{,}{4}{,}{6}\right\}{,}\left\{{3}{,}{5}{,}{6}\right\}{,}\left\{{4}{,}{5}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{5}{,}{6}\right\}\right]$ (5)

References

 James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.