An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\geq 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and no member of a specific family of $3$-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified $3$-connected minor. We also prove that if $M$ has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.
@article{10_37236_11399,
author = {George Drummond and Charles Semple},
title = {A splitter theorem for elastic elements in 3-connected matroids},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11399},
zbl = {1520.05022},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11399/}
}
TY - JOUR
AU - George Drummond
AU - Charles Semple
TI - A splitter theorem for elastic elements in 3-connected matroids
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11399/
DO - 10.37236/11399
ID - 10_37236_11399
ER -
%0 Journal Article
%A George Drummond
%A Charles Semple
%T A splitter theorem for elastic elements in 3-connected matroids
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11399/
%R 10.37236/11399
%F 10_37236_11399
George Drummond; Charles Semple. A splitter theorem for elastic elements in 3-connected matroids. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/11399
