# Quasiregular Matroids

## Sandra Kingan, College/CUNY

## Date and time: 2pm, Friday, September 14, 2018

## Place: CUNY Graduate Center, Rm 4419

Regular matroids are binary matroids with no minors isomorphic to the
Fano matroid or its dual. The Fano matroid is the binary projective
plane PG(2, 2). Seymour proved that 3-connected regular matroids are
either graphs, cographs, or a special 10-element matroid R10 called a
splitter, or else can be decomposed along a non-minimal exact
3-separation induced by another special matroid R12 called a
3-decomposer. Quasiregular matroids are binary matroids with no minor
isomorphic to E4, where E4 is a 10-element rank 5 self-dual binary
matroid. The class of quasiregular matroids properly contains the
class of regular matroids. In a paper that just appeared in the
Electronic Journal of Combinatorics (Vol. 25, Issue 3 (2018)), we
decomposed the class of quasiregular matroids in a manner similar to
regular matroids by showing that quasiregular matroids are either
graphs, cographs, or deletion-minors of PG(3,2), R17 or M12 or else
can be decomposed along a non-minimal exact 3-separation induced by
R12, P9, or its dual. The matroid M12 is a splitter and PG(3,2) and
R17 are the maximal 3-connected rank 4 and 5 members, respectively.
The class of quasiregular matroids is one of very few excluded minor
classes whose members have been described. After Seymour's
decomposition of regular matroids, the Fano dual became the starting
point for the analysis of binary non-graphic and non-cographic
matroids. As a consequence of the decomposition of quasiregular
matroids the starting point for the analysis of binary non-graphic and
non-cographic matroids is E4.