Subdivisions in apex graphs

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 $K_{{}_5}$

30 pages, 3 figures, submitted on June 26th 2010