Fitch Graph Completion

TL;DR

This paper characterizes Fitch-satisfiable graphs related to horizontal gene transfer, provides a polynomial-time recognition algorithm, and proves that maximizing confidence edge-weights is NP-hard.

Contribution

It introduces a characterization and recognition algorithm for Fitch-satisfiable graphs and establishes NP-hardness of the maximum confidence Fitch graph problem.

Findings

Characterization of Fitch-satisfiable graphs

Polynomial-time recognition algorithm for Fitch graphs

NP-hardness of maximizing total confidence edge-weights

Abstract

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.