The circuits of a matroid are its minimal dependent sets. Given a matroid, the Circuits command returns the circuits of that matroid as a list of sets.

If the circuits of this matroid haven't been computed before, they are computed by this command and stored for any future computations that use the circuits.

$\mathrm{with}\left(\mathrm{Matroids}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,1\,1\,0\,0\,0\,0\right]\,\left[0\,0\,0\,1\,1\,1\,0\right]\,\left[1\,0\,0\,1\,0\,0\,0\right]\,\left[0\,1\,0\,0\,1\,0\,0\right]\,\left[0\,0\,1\,0\,0\,2\,0\right]\right]\right)$

$A≔\left[\begin{array}{ccccccc}1& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 1& 1& 0\\ 1& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 2& 0\end{array}\right]$

$M≔\mathrm{Matroid}\left(A\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{ccccccc}1& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 1& 1& 0\\ 1& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 2& 0\end{array}\right]\end{array}⟩$

$C≔\mathrm{Circuits}\left(M\right)$

$C≔\left[\left\{7\right\}\,\left\{1\,2\,4\,5\right\}\right]$

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(\left\{\left\{a\,x\right\}\,\left\{a\,y\right\}\,\left\{b\,x\right\}\,\left\{b\,y\right\}\,\left\{c\,x\right\}\,\left\{c\,y\right\}\right\}\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$M≔\mathrm{Matroid}\left(G\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; graphic\; matroi\ⅆ\; on\; th\ⅇ\; graph:}\\ \mathrm{PLOT}\left(\mathrm{...}\right)\end{array}⟩$

$C≔\mathrm{Circuits}\left(M\right)$

$C≔\left[\left\{''a\_x''\,''a\_y''\,''b\_x''\,''b\_y''\right\}\,\left\{''a\_x''\,''a\_y''\,''c\_x''\,''c\_y''\right\}\,\left\{''b\_x''\,''b\_y''\,''c\_x''\,''c\_y''\right\}\right]$

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