A splitter theorem on 3-connected matroids and graphs

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 $k:=\mathrm{\mid }V(G)\mathrm{\mid }-\mathrm{\mid }V(H)\mathrm{\mid }\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$\mathrm{.}Thentherearepairwisedisjointsets$X_1,\dots,X_n\subseteq E(G)$suchthateach$G/X_i$isa$3$-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 $k=3$).