Independent paths and K5-subdivisions

A well known theorem of Kuratowski states that a graph is planar iff it contains no subdivision of K5 or K3,3. Seymour conjectured in 1977 that every 5-connected nonplanar graph contains a subdivision of K5. In this paper, we prove several results about independent paths (no vertex of a path is internal to another), which are then used to prove Seymour’s conjecture for two classes of graphs. These results will be used in a subsequent paper to prove Seymour’s conjecture for graphs containing K − 4, which is a step in a program to approach Seymour’s conjecture