Editing Matroid (section)

===Minors===
{{main|Matroid minor}}
If ''M'' is a matroid with element set ''E'', and ''S'' is a subset of ''E'', the ''restriction'' of ''M'' to ''S'', written ''M''&nbsp;|''S'', is the matroid on the set ''S'' whose independent sets are the independent sets of ''M'' that are contained in ''S''.  Its circuits are the circuits of ''M'' that are contained in ''S'' and its rank function is that of ''M'' restricted to subsets of ''S''.

In linear algebra, this corresponds to restricting to the subspace generated by the vectors in ''S''.  Equivalently if ''T'' = ''M''−''S'' this may be termed the ''deletion'' of ''T'', written ''M''\''T'' or ''M''−''T''. The submatroids of ''M'' are precisely the results of a sequence of deletions: the order is irrelevant.<ref name=Whi86131>{{harvp|White|1986|p=131}}</ref><ref name=Whi86224>{{harvp|White|1986|p=224}}</ref>

The dual operation of restriction is contraction.<ref name=Whi866139>{{harvp|White|1986|p=139}}</ref> If ''T'' is a subset of ''E'', the ''contraction'' of ''M'' by ''T'', written ''M''/''T'', is the matroid on the underlying set ''E &minus; T'' whose rank function is <math>r'(A) = r(A \cup T) - r(T)</math>.<ref name=Whi86140>{{harvp|White|1986|p=140}}</ref>  In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in ''T'', together with the images of the vectors in ''E - T''.

A matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations is called a [[matroid minor|minor]] of ''M''.<ref name=Whi86224/><ref name=Whi86150>{{harvp|White|1986|p=150}}</ref>  We say ''M'' ''contains'' ''N'' ''as a minor''. Many important families of matroids may be characterized by the [[minimal element|minor-minimal]] matroids that do not belong to the family; these are called ''forbidden'' or ''excluded minors''.<ref name=Whi861467>{{harvp|White|1986|pp=146–147}}</ref>