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.

翻译：由标签的根植树产生的二进制关系在数学生物学中作为正式的进化关系模式发挥着进口作用。(平衡化)惠誉关系正式将美元作为由至少一个水平转移事件分离的基因配对。作为一个自然的概括化,我们考虑对齐化的惠誉地图,即对称地图$\varepsilon$,为每对脊椎指定一组颜色,以美元为x$,然后用一棵树美元解释,其边缘以颜色子组为标签。如果而且只有美元出现在美元和美元之间的独一路径上的标签上,色(美元)。我们首先对单色性案例进行另外的描述,然后对每对每对一对一对脊椎的颜色配色进行定性,然后用一组四重奏的地图的兼容性来描述。我们展示了对调化的Flicklicklon(x,ymaldrial Flal-lal-lorma) 图表的完整分数级图的识别度,自1个限制案例以来,其等级图的正态形式就成为了。