  Interacting RNA complexes are studied via bicellular maps using a filtration
via their topological genus. Our main result is a new bijection for RNA-RNA
interaction structures and linear time uniform sampling algorithm for RNA
complexes of fixed topological genus. The bijection allows to either reduce the
topological genus of a bicellular map directly, or to lose connectivity by
decomposing the complex into a pair of single stranded RNA structures. Our main
result is proved bijectively. It provides an explicit algorithm of how to
rewire the corresponding complexes and an unambiguous decomposition grammar.
Using the concept of genus induction, we construct bicellular maps of fixed
topological genus $g$ uniformly in linear time. We present various statistics
on these topological RNA complexes and compare our findings with biological
complexes. Furthermore we show how to construct loop-energy based complexes
using our decomposition grammar.
