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The Kelmans-Seymour conjecture I: special separations
Seymour and, independently, Kelmans conjectured in the 1970s that every
5-connected nonplanar graph contains a subdivision of . This conjecture
was proved by Ma and Yu for graphs containing , 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