Deletable edges in 3-connected graphs and their applications

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has no $H$-deletable edges, then there exists a sequence of simple 3-connected graphs $G_0, \dots , G_k$ with no $H$-deletable edges such that $G_0\cong H$, $G_k= G$, and for $1 \le i \le k$ one of three possibilities holds: $G_{i-1}= G_i/f$; $G_{i-1}=G_i/f \backslash e$ where $e$ and $f$ are incident to a degree 3 vertex in $G_i$; or $G_{i-1}=G_i-w$ where $w$ is a degree $3$ vertex in $G_i$. Several applications are given including a graph theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph. Halin proved that if $G$ is a minimally $3$-connected graph on $n\ge 8$ vertices, then $|E(G)|\le 3n-9$ and equality holds if and only if $G\cong K_{3, n-3}$. We give a different proof of Halin's result and extend it by identifying the minimally 3-connected infinite family of graphs with $|E(G)|=3n-10$.