Subdivisions in planar graphs

Abstract

Given four distinct vertices in a 4-connected planar graph G, we characterize when the graph G contains a K4 -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 no K4-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. © 1998 Academic Press.