The Kelmans-Seymour conjecture III: 3-vertices in $K_4^{-}$

Abstract

Let $G$ be a 5-connected nonplanar graph and let $x_1,x_2,y_1,y_2\in V(G)$ be distinct, such that $G[\{x_1,x_2,y_1,y_2\}]\cong K_4^{-}$ and $y_1{y}_2\notin E(G)$. We show that one of the following holds: $G-x_1$ contains $K_4^{-}$, or $G$ contains a $K_4^{-}$ in which $x_1$ is of degree 2, or $G$ contains a $T{K}_5$ in which $x_1$ is not a branch vertex, or $\{x_2,y_1,y_2\}$ may be chosen so that for any distinct $z_0,z_1\in N(x_1)-\{x_2,y_1,y_2\}$, $G-\{x_{1}v:v\notin \{z_0,z_1,x_2,y_1,y_2\}\}$ contains $T{K}_5$. This result will be used to prove the Kelmans-Seymour conjecture.