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 but FPT in general. 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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广义 Fitch 图 III：由无根边标记树解释的对称 Fitch 映射和对称二元关系集

源自标记的有根树的二元关系作为进化关系的正式模型在数学生物学中发挥着重要作用。（对称的）Fitch 关系将异种学形式化为由至少一个水平转移事件分隔的基因对。作为自然概括，我们考虑对称 Fitch 映射，即对称映射 $\varepsilon$，它为 $X$ 中的每对顶点分配一个颜色子集，并且可以通过具有边的树 $T$ 来解释用颜色子集标记，即颜色 $m$ 出现在 $\varepsilon(x,y)$ 中当且仅当 $m$ 出现在沿着 $x$ 和 $y$ 之间的唯一路径的标签中时美元。我们首先给出单色情况的另一种表征，然后根据一组特定四重奏的兼容性给出对称 Fitch 映射的表征。我们表明对称 Fitch 映射的识别是 NP 完全的，但通常是 FPT。在 $|\varepsilon(x,y)|\leq 1$ 的受限情况下，问题变成多项式，因为这样的映射与单色 Fitch 映射的类重合，其图形表示恰好形成完整的多部分图类。