Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^{-}$, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture.

西摩（Seymour）和凯尔曼斯（Kelmans）独立地推测，在1970年代，每个5连通的非平面图都包含一个细分 ${ķ}_5$。Ma和Yu对包含${ķ}_4^{-}$，证明它们的一个重要步骤是处理图形中带有平面侧的5个分隔。为了为所有图建立Kelmans-Seymour猜想，我们需要考虑具有较少限制结构的5分离和6分离。本文的目的是处理特殊的5分离和6分离，包括那些带有顶点的分离。结果将用于后续论文中以证明Kelmans-Seymour猜想。