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

Fitch graphs G = (X,E) are di-graphs that are explained by {⊗, 1}-edge-labeled rooted trees with leaf set X: there is an arc xy ∈ E if and only if the unique path in T that connects the least common ancestor 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 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 ε that assigns to each arc xy a unique label ε(x, y) ∈ M ∪ {⊗}, where M denotes an arbitrary set of symbols. A di-graph (K|X|, ε) is a generalized Fitch graph if there is an M ∪ {⊗}-edge-labeled tree (T, λ) that can explain (K|X|, ε). We provide a simple characterization of generalized Fitch graphs (K|X|, ε) and give an O(|X|2)-time algorithm for their recognition as well as for the reconstruction of a least resolved phylogenetic tree that explains (K|X|, ε).