The Kelmans-Seymour conjecture II: 2-vertices in $K_4^{-}$

Abstract

We use $K_4^{-}$ to denote the graph obtained from $K_4$ by removing an edge, and use $T{K}_5$ to denote a subdivision of $K_5$. Let $G$ be a 5-connected nonplanar graph and $\{x_1,x_2,y_1,y_2\}\subseteq V(G)$ such that $G[\{x_1,x_2,$ $y_1,y_2\}]\cong K_4^{-}$ with $y_1{y}_2\notin E(G)$. Let $w_1,w_2,w_3\in N(y_2)-\{x_1,x_2\}$ be distinct. We show that $G$ contains a $T{K}_5$ in which $y_2$ is not a branch vertex, or $G-y_2$ contains $K_4^{-}$, or $G$ has a special 5-separation, or $G-\{y_{2}v:v\notin \{w_1,w_2,w_3,x_1,x_2\}\}$ contains $T{K}_5$.