A matroid $\mathrm{M1}$ is a minor of another matroid $\mathrm{M2}$ if $\mathrm{M1}$ can be obtained from $\mathrm{M2}$ via a sequence of deletion and contraction operations. This procedure determines if $\mathrm{M1}$ is a minor of $\mathrm{M2}$.

In some cases, this question can be decided quickly, e.g., if $\mathrm{M1}$ has more entries in its ground set than $\mathrm{M2}$, the answer is no. Otherwise, this command enters into a loop that can take quite a long time to run. By default, Maple will issue an error message if it computes that the loop may take more than $2^{20}$ iterations. To use a different limit for the current call, you can supply the option $\mathrm{maxiterations}=m$, where $m$ is either a positive integer or the symbol $\mathrm{\infty }$.

By default, if $\mathrm{M1}$ is not a minor of $\mathrm{M2}$, this command returns the value $\mathrm{false}$. If $\mathrm{M1}$ is indeed a minor of $\mathrm{M2}$, it returns a sequence of three values: the constant $\mathrm{true}$ and two subsets $C$ and $\mathrm{D}$ of the ground set of $\mathrm{M2}$ such that $\mathrm{M1}$ is isomorphic to $\left(\mathrm{M2}\setminus \mathrm{D}\right)/C$. The output can be modified using an option of the form $\mathrm{output}=o$.

If you pass the option $\mathrm{output}=\mathrm{boolean}$, the command returns just $\mathrm{true}$ or $\mathrm{false}$.

If you pass the option $\mathrm{output}=\mathrm{contractions}$, the command returns just $C$ if $\mathrm{M1}$ is a minor of $\mathrm{M2}$, and the symbol $\mathrm{FAIL}$ otherwise.

If you pass the option $\mathrm{output}=\mathrm{deletions}$, the command returns just $\mathrm{D}$ if $\mathrm{M1}$ is a minor of $\mathrm{M2}$, and the symbol $\mathrm{FAIL}$ otherwise.

If you pass the option $\mathrm{output}=l$, where $l$ is a list consisting of any of the values $\mathrm{boolean}$, $\mathrm{contractions}$, and $\mathrm{deletions}$, the command returns an expression sequence of the corresponding outputs in the given order.

If you pass the option $\mathrm{output}=\mathrm{default}$, the command behaves as if the option had not been passed.

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

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

$\mathrm{M1}≔\mathrm{UniformMatroid}\left(1\,2\right)$

$\mathrm{M1}≔⟨\mathrm{th\ⅇ\; uniform\; matroi\ⅆ\; of\; rank\; 1\; on\; 2\; \ⅇl\ⅇm\ⅇnts}⟩$

$\mathrm{M2}≔\mathrm{Fano}\left(\right)$

$\mathrm{M2}≔⟨\mathrm{a\; matroi\ⅆ\; on\; 7\; \ⅇl\ⅇm\ⅇnts\; with\; 14\; circuits}⟩$

$\mathrm{IsMinorOf}\left(\mathrm{M1}\,\mathrm{M2}\right)$

$\mathrm{true},\left\{4\right\},\left\{3\,5\,6\,7\right\}$

$\mathrm{M3}≔\mathrm{Contraction}\left(\mathrm{Deletion}\left(\mathrm{M2}\,\left\{3\,5\,6\,7\right\}\right)\,\left\{4\right\}\right)$

$\mathrm{M3}≔⟨\mathrm{th\ⅇ\; uniform\; matroi\ⅆ\; of\; rank\; 1\; on\; 2\; \ⅇl\ⅇm\ⅇnts}⟩$

$\mathrm{AreIsomorphic}\left(\mathrm{M1}\,\mathrm{M3}\right)$

$\mathrm{true}$

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