从单细胞脂肪图到树木
From unicellular fatgraphs to trees
摘要：
在本文中，我们研究的最小数量的逆转需要将一个单细胞胖图转化为一棵树。我们考虑作用于边界分量的反转，具有作为顶点的胶合、切片或半翻转的自然解释。我们的主要结果是一个表达式的最小数量的反转需要转换一个单细胞胖图的平面树。该表达式包含了胖图的欧拉亏格和一个额外的参数，该参数计算了胖图分解中某些可定向块的数量。在这个过程中，我们得出一个建设性的证明，如何分解成较小的相同类型的或平凡的胖图，由一个单一的丝带的非定向，不可约，单细胞胖图。此外，我们还提供了一个详细的分析如何逆转影响底层胖图的组件结构。我们的结果推广了符号置换的反转距离的Hannenhalli-Pevzner公式。
英文摘要：
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.
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