On the frequency of 3-connected subgraphs of planar graphs

The concept of dependence of subgraphs of a plane graph is defined, as a measure of how much they overlap. It is shown that if M is a 3 -connected plane graph, then the number of copies of M in a plane graph which are dependent on a given copy is bounded above by a constant c ( M ). The number of copies of M in any n -vertex plane graph is at most nc ( m ).