  Tree decompositions were developed by Robertson and Seymour. Since then
algorithms have been developed to solve intractable problems efficiently for
graphs of bounded treewidth. In this paper we extend tree decompositions to
allow cycles to exist in the decomposition graph; we call these new
decompositions plane decompositions because we require that the decomposition
graph be planar. First, we give some background material about tree
decompositions and an overview of algorithms both for decompositions and for
approximations of planar graphs. Then, we give our plane decomposition
definition and an algorithm that uses this decomposition to approximate the
size of the maximum independent set of the underlying graph in polynomial time.
