  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.
