  In 2009, Brown gave a set of conditions which when satisfied imply that a
Feynman integral evaluates to a multiple zeta value. One of these conditions is
called reducibility, which loosely says there is an order of integration for
the Feynman integral for which Brown's techniques will succeed. Reducibility
can be abstracted away from the Feynman integral to just being a condition on
two polynomials, the first and second Symanzik polynomials. These polynomials
can be defined from graphs, and thus reducibility is a property of graphs. We
prove that for a fixed number of external momenta and no masses, reducibility
is graph minor closed, correcting the previously claimed proofs of this fact. A
computational study of reducibility was undertaken by Bogner and L\"{u}ders who
found that for graphs with $4$-on-shell momenta and no masses, $K_{4}$ with
momenta on each vertex is a forbidden minor. We add to this and find that when
we restrict to graphs with four on-shell external momenta the following graphs
are forbidden minors: $K_{4}$ with momenta on each vertex, $W_{4}$ with
external momenta on the rim vertices, $K_{2,4}$ with external momenta on the
large side of the bipartition, and one other graph. We do not expect that these
minors characterize reducibility, so instead we give structural
characterizations of the graphs not containing subsets of these minors. We
characterize graphs not containing a rooted $K_{4}$ or rooted $W_{4}$ minor,
graphs not containing rooted $K_{4}$ or rooted $W_{4}$ or rooted $K_{2,4}$
minors, and also a characterization of graphs not containing all of the known
forbidden minors. Some comments are made on graphs not containing $K_{3,4}$,
$K_{6}$ or a graph related to Wagner's graph as a minor.
