Generalized Fitch Graphs: Edge-labeled Graphs that are explained by Edge-labeled Trees

TL;DR

This paper extends Fitch graphs to a more general form by allowing arbitrary edge labels and provides efficient algorithms for recognizing these graphs and reconstructing their explaining trees.

Contribution

It introduces generalized Fitch graphs with arbitrary labels and offers a simple characterization and an $O(n^2)$ recognition and reconstruction algorithm.

Findings

Characterization of generalized Fitch graphs

Efficient $O(n^2)$ recognition algorithm

Unique least resolved explaining tree

Abstract

Fitch graphs $G=(X,E)$ are di-graphs that are explained by $\{\otimes,1\}$-edge-labeled rooted trees with leaf set $X$: there is an arc $xy\in E$ if and only if the unique path in $T$ that connects the least common ancestor $\textrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label $1$. In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\textrm{lca}(x,y)$ to $y$. In this contribution, we generalize the concept of xenology and Fitch graphs and consider complete di-graphs $K_{|X|}$ with vertex set $X$ and a map $\epsilon$ that assigns to each arc $xy$ a unique label $\epsilon(x,y)\in M\cup \{\otimes\}$, where $M$ denotes an arbitrary set of symbols. A di-graph $(K_{|X|},\epsilon)$ is a generalized Fitch graph if there is an $M\cup \{\otimes\}$-edge-labeled tree $(T,\lambda)$ that can explain $(K_{|X|},\epsilon)$. We provide a simple characterization of generalized Fitch graphs $(K_{|X|},\epsilon)$ and give an $O(|X|^2)$-time algorithm for their recognition as well as for the reconstruction of the unique least resolved phylogenetic tree that explains $(K_{|X|},\epsilon)$.