Plane Decompositions as Tools for Approximation
  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. 
Melanie J. Agnew and Christopher M. Homan