Horizontal gene transfer is an important contributor to evolution. According to Walter M.\ Fitch, two genes are xenologs if they are separated by at least one HGT. More formally, the directed Fitch graph has a set of genes is its vertices, and directed edges $(x,y)$ for all pairs of genes $x$ and $y$ for which $y$ has been horizontally transferred at least once since it diverged from the last common ancestor of $x$ and $y$. Subgraphs of Fitch graphs can be inferred by comparative sequence analysis. In many cases, however, only partial knowledge about the ``full'' Fitch graph can be obtained. Here, we characterize Fitch-satisfiable graphs that can be extended to a biologically feasible ``full'' Fitch graph and derive a simple polynomial-time recognition algorithm. We then proceed to showing that finding the Fitch graphs with total maximum (confidence) edge-weights is an NP-hard problem.

翻译：水平基因转移是进化的重要推动因素。根据Walter M. Fitch的定义，两个基因如果被至少一次水平基因转移事件所分隔，则它们互为异源同源基因。更正式地，有向菲奇图的顶点集由一组基因构成，且对于任意一对基因$x$和$y$，如果$y$在与$x$的最后共同祖先分化后至少经历了一次水平转移，则图中存在有向边$(x, y)$。通过比较序列分析可以推断出菲奇图的子图。然而，在许多情况下只能获得关于“完整”菲奇图的局部信息。本文描述了可扩展为生物学可行“完整”菲奇图的菲奇可满足图，并推导出简单的多项式时间识别算法。我们进一步证明，寻找具有最大总（置信）边权重的菲奇图是NP难问题。