Estimates of AGMON-MIRANDA type for solutions of the DIRICHLET problem in plane domains with corners

It is proved that any edge of a 4-connected non-planar graph G of order at least 6 lies in a subdivision of K_3,3 in G. For any 3-connected non-planar graph G of order at least 6 we show that G contains at most four edges which belong to no subdivisions of K_3,3 in G.