From unicellular fatgraphs to trees

TL;DR

This paper derives a formula for the minimum reversals needed to convert unicellular fatgraphs into trees, generalizing known permutation reversal results and providing a constructive decomposition method.

Contribution

It introduces a new expression for reversal distance involving Euler genus and orientable blocks, extending the Hannenhalli-Pevzner formula to fatgraphs.

Findings

Derived a formula for reversal distance involving Euler genus and orientable blocks.

Provided a constructive decomposition method for fatgraphs.

Analyzed how reversals impact the component structure of fatgraphs.

Abstract

In this paper we study the minimum number of reversals needed to transform a unicellular fatgraph into a tree. We consider reversals acting on boundary components, having the natural interpretation as gluing, slicing or half-flipping of vertices. Our main result is an expression for the minimum number of reversals needed to transform a unicellular fatgraph to a plane tree. The expression involves the Euler genus of the fatgraph and an additional parameter, which counts the number of certain orientable blocks in the decomposition of the fatgraph. In the process we derive a constructive proof of how to decompose non-orientable, irreducible, unicellular fatgraphs into smaller fatgraphs of the same type or trivial fatgraphs, consisting of a single ribbon. We furthermore provide a detailed analysis how reversals affect the component-structure of the underlying fatgraphs. Our results generalize the Hannenhalli-Pevzner formula for the reversal distance of signed permutations.