The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M0,...,Mn
Nearby in the stack
of 3-connected matroids with
M0≅N
,
Mn=M
and for
i∈{1,...,n}
,
Mi
is a single-element extension or coextension of
Mi−1
. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism,
M
starting with
N
and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of the matroids involved are
r(M)
). Moreover, if two consecutive single-element extensions by elements
{e,f}
are followed by a coextension by element
g
, then
{e,f,g}
form a triad in the resulting matroid. Using the Strong Splitter Theorem, we make progress toward the problem of determining the almost-regular matroids [6, 15.9.8]. Find all 3-connected non-regular matroids such that, for all
e
, either
M\e
or
M/e
is regular. In [4] we determined the binary almost-regular matroids with at least one regular element (an element such that both
M\e
and
M/e
is regular) by characterizing the class of binary almost-regular matroids with no minor isomorphic to one particular matroid that we called
E5
. As a consequence of the Strong Splitter Theorem we can determine the class of binary matroids with an