  In this paper we present a combinatorial proof of a relation between the
generating functions of unicellular and bicellular maps. This relation is a
consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it
can be proved using representation theory of the symmetric group. Here we give
a bijective proof by rewiring unicellular maps of topological genus $(g+1)$
into bicellular maps of genus $g$ and pairs of unicellular maps of lower
topological genera. Our result has immediate consequences for the folding of
RNA interaction structures, since the time complexity of folding the
transformed structure is $O((n+m)^5)$, where $n,m$ are the lengths of the
respective backbones, while the folding of the original structure has $O(n^6)$
time complexity, where $n$ is the length of the longer sequence.
