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

TL;DR

This paper investigates the structure of 5-connected nonplanar graphs containing a specific subgraph, establishing conditions under which certain subdivisions or configurations must exist, advancing the understanding of the Kelmans-Seymour conjecture.

Contribution

It proves new structural results about 5-connected nonplanar graphs with a $K_4^-$ subgraph, identifying conditions for the existence of $TK_5$ subdivisions or special separations.

Findings

Existence of a $TK_5$ with specific properties under given conditions

Presence of $K_4^-$ in $G-y_2$ or a special 5-separation

Identification of configurations preventing $TK_5$ formation

Abstract

We use $K_4^-$ to denote the graph obtained from $K_4$ by removing an edge, and use $TK_5$ 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 $y_1y_2\notin E(G)$. Let $w_1,w_2,w_3\in N(y_2)-\{x_1,x_2\}$ be distinct. We show that $G$ contains a $TK_5$ in which $y_2$ is not a branch vertex, or $G-y_2$ contains $K_4^-$, or $G$ has a special 5-separation, or $G-\{y_2v:v\notin \{w_1,w_2,w_3,x_1,x_2\}\}$ contains $TK_5$.