On recognizing graphs representing Persistent Perfect Phylogenies

TL;DR

This paper introduces a polynomial-time algorithm for recognizing graphs that represent Persistent Perfect Phylogenies, extending the classic perfect phylogeny model to account for character loss, and focusing on maximal graphs.

Contribution

It provides the first polynomial-time recognition algorithm for a class of graphs representing Persistent Perfect Phylogenies, bridging the gap between simple and complex phylogeny models.

Findings

Polynomial-time algorithm for maximal graphs

Recognition relies solely on graph properties

Bridges the gap between simple and NP-hard phylogeny models

Abstract

The Persistent Perfect phylogeny, also known as Dollo-1, has been introduced as a generalization of the well-known perfect phylogenetic model for binary characters to deal with the potential loss of characters. The problem of deciding the existence of a Persistent Perfect phylogeny can be reduced to the one of recognizing a class of bipartite graphs whose nodes are species and characters. Thus an interesting question is solving directly the problem of recognizing such graphs. We present a polynomial-time algorithm for deciding Persistent Perfect phylogeny existence in maximal graphs, where no character's species set is contained within another character's species set. Our solution, that relies only on graph properties, narrows the gap between the linear-time simple algorithm for Perfect Phylogeny and the NP-hardness results for the Dollo-$k$ phylogeny with $k>1$.