Matroid minor

Remarks

If $J$ is a maximal independent subset of $X$, then a set $I \subseteq E \setminus X$ is independent in $M / X$ if and only if $I \cup J \in \mathcal{I}$. Deletion and contraction commute, and $M / X = (M^* \setminus X)^*$, where $M^*$ is the dual matroid. Many authors call $N$ a minor when it is isomorphic to a matroid of the form in clause 3. For the graphic matroid of a graph, deleting or contracting an edge gives the graphic matroid of the graph with that edge deleted or contracted.

Example: let $M = U_{2,4}$, the matroid on $\set{1, 2, 3, 4}$ whose independent sets are the subsets with at most $2$ elements. Then $M \setminus \set{4} = U_{2,3}$, and $M / \set{4} = U_{1,3}$, since $r(A \cup \set{4}) - 1 = \min(\abs{A} + 1, 2) - 1 = \min(\abs{A}, 1)$.

Non-example: $U_{2,4}$ is not a minor of $U_{1,5}$, because deletion and contraction never raise the rank, and $U_{2,4}$ has rank $2$ while $U_{1,5}$ has rank $1$.

A theorem of Tutte states that a matroid is representable over the field with two elements if and only if it has no minor isomorphic to $U_{2,4}$.