Subdivisions in Planar Graphs

Given four distinct vertices in a 4-connected planar graphG, we characterize when the graphGcontains aK4-subdivision with the given vertices as its degree three vertices. This result implies the following conjecture of Robertson and Thomas: a 5-connected planar graph has noK4-subdivision with specified degree three vertices, if and only if the four specified vertices are contained in a facial cycle in the unique plane embedding of the graph.