On recognizing graphs representing Persistent Perfect Phylogenies

2025-07-24

AI Summary

This paper addresses the existence problem of Persistent Perfect Phylogenies (PPPs) on species–character bipartite graphs under the “maximum graph” constraint—i.e., no two character state sets are inclusion-related. We present the first purely graph-theoretic polynomial-time algorithm for PPP existence testing. Our method leverages structural analysis of the bipartite graph and encodes character-set inclusion relations directly into auxiliary graph representations, bypassing exhaustive character-state enumeration or dynamic programming. This approach bridges a fundamental theoretical gap between classical Perfect Phylogeny—which admits linear-time solutions—and the Dollo-*k* model for *k* > 1, which is NP-hard. Crucially, our algorithm preserves biological interpretability while substantially improving computational tractability for complex evolutionary models. It establishes a rigorous, efficient graph-theoretic framework for phylogenetic inference, offering both theoretical insight and practical utility in computational phylogenetics.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $k>1$.

Problem

• Recognizing graphs representing Persistent Perfect Phylogenies
• Deciding existence of Persistent Perfect phylogeny in maximal graphs
• Developing polynomial-time algorithm for Persistent Perfect phylogeny

Innovation

• Polynomial-time algorithm for Persistent Perfect Phylogeny
• Focuses on maximal graphs with unique character sets
• Uses graph properties to simplify complex phylogeny