A splitter theorem on 3-connected matroids and graphs

TL;DR

This paper proves a splitter theorem for 3-connected graphs and matroids, identifying disjoint sets that preserve connectivity and minors, extending previous results for small differences in vertex counts.

Contribution

It generalizes splitter theorems for 3-connected graphs and matroids to larger differences in vertex counts, providing a new structural decomposition method.

Findings

Identifies disjoint sets preserving 3-connectivity and minors

Extends previous splitter theorems for larger vertex differences

Provides a structural decomposition involving triangles and singleton sets

Abstract

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$. Then there are pairwise disjoint sets $X_1,\dots,X_n\subseteq E(G)$ such that each $G/X_i$ is a $3$-connected graph with an $H$-minor, each $X_i$ is a singleton set or the edge set of a triangle of $G$ with $3$ degree-$3$ vertices and $X_1\cup\cdots\cup X_n$ contains no edge sets of circuits of $G$ other than the $X_i$'s. This result extends previous ones of Whittle (for $k=1,2$) and Costalonga (for $k=3$).