Excluded minors are almost fragile

Abstract

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb{P}$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded relative to $N$, or, up to replacing $M$ by a $\Delta$-$Y$-equivalent excluded minor, we can choose a pair of elements $\{a,b\}$ such that either $M\backslash \{a,b\}$ is $N$-fragile, or $M^{\ast }\backslash \{a,b\}$ is $N^{\ast }$-fragile.

arXiv comment53 pages

References in corpus (4)

• Confinement of matroid representations to subsets of partial fields
• Lifts of matroid representations over partial fields
• $N$-detachable pairs in 3-connected matroids I: unveiling $X$
• $N$-detachable pairs in 3-connected matroids III: the theorem

Cited by in corpus (5)

• $N$-detachable pairs in 3-connected matroids I: unveiling $X$
• The excluded minors for 2- and 3-regular matroids
• Computing excluded minors for classes of matroids representable over partial fields
• Excluded minors are almost fragile II: essential elements
• The excluded minors for GF(5)-representable matroids on ten elements