A bijection between unicellular and bicellular maps

TL;DR

This paper provides a bijective combinatorial proof linking unicellular and bicellular maps, with implications for RNA structure folding complexity.

Contribution

It introduces a bijective proof connecting unicellular and bicellular maps, offering new insights into their relationship and applications in RNA folding algorithms.

Findings

Established a bijection between unicellular and bicellular maps of related genera.

Derived a complexity reduction from $O(n^6)$ to $O((n+m)^5)$ for RNA folding algorithms.

Provided combinatorial and algebraic proofs of the map relation.

Abstract

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.