Subdivisions in apex graphs

TL;DR

This paper proves that 5-connected nonplanar apex graphs containing a $K_{2,3}$ subgraph necessarily have a subdivided $K_5$, advancing understanding of the Kelmans-Seymour conjecture.

Contribution

It establishes that such apex graphs with a $K_{2,3}$ subgraph contain a subdivided $K_5$, filling a gap in the plan towards the conjecture.

Findings

Proves the existence of subdivided $K_5$ in the specified graphs.

Extends previous results from $K^-_4$ to $K_{2,3}$.

Supports the plan to resolve the Kelmans-Seymour conjecture.

Abstract

The Kelmans-Seymour conjecture states that the 5-connected nonplanar graphs contain a subdivided $K_{_5}$. Certain questions of Mader propose a "plan" towards a possible resolution of this conjecture. One part of this plan is to show that a 5-connected nonplanar graph containing $K^-_{_4}$ or $K_{_{2,3}}$ as a subgraph has a subdivided $K_{_5}$. Recently, Ma and Yu showed that a 5-connected nonplanar graph containing $K^-_{_4}$ as a subgraph has a subdivided $K_{_5}$. We take interest in $K_{_{2,3}}$ and prove that a 5-connected nonplanar apex graph containing $K_{_{2,3}}$ as a subgraph has a subdivided $K_{_5}$