广义Fitch图III：对称化Fitch映射与由无根边标记树解释的对称二元关系集
Generalized Fitch Graphs III: Symmetrized Fitch maps and Sets of Symmetric Binary Relations that are explained by Unrooted Edge-labeled Trees
AI总结：
本文研究对称化Fitch映射的刻画与识别，给出基于四元组兼容性的新刻画，并证明一般识别是NP完全的，而限制每对顶点至多一种颜色时退化为多项式可解的单色Fitch映射。
摘要：
由标记的有根树导出的二元关系在数学生物学中作为进化关系的正式模型起着重要作用。（对称化的）Fitch关系将异种学形式化为由至少一个水平基因转移事件分隔的基因对。作为自然推广，我们考虑对称化的Fitch映射，即对称映射$\varepsilon$，它为$X$中每对顶点分配一个颜色子集，并且可以由树$T$解释，其中边标记为颜色子集，使得颜色$m$出现在$\varepsilon(x,y)$中当且仅当$m$出现在$T$中$x$和$y$之间唯一路径上的某个标签中。我们首先给出单色情况的另一种刻画，然后基于一组特定四元组的兼容性给出对称化Fitch映射的刻画。我们证明识别对称化Fitch映射是NP完全的。在限制情况$|\varepsilon(x,y)|\leq 1$下，问题变为多项式可解的，因为此类映射与单色Fitch映射类一致，其图表示恰好构成完全多部图类。
英文摘要：
Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.
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