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 $y_1y_2\notin E(G)$. 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 $TK_5$ in which $x_1$ is not a branch vertex, or $\{x_2,y_1,y_2\}$ may be chosen so that for any distinct $z_0, z_1\in N(x_1)-\{x_2,y_1,y_2\}$, $G-\{x_1v:v\notin \{z_0, z_1,x_2, y_1,y_2\}\}$ contains $TK_5$. This result will be used to prove the Kelmans-Seymour conjecture.