Recognizing Leaf Powers and Pairwise Compatibility Graphs is NP-Complete

TL;DR

This paper proves that recognizing leaf powers and pairwise compatibility graphs is NP-complete, highlighting the computational difficulty of identifying these graph classes, which model phylogenetic trees.

Contribution

It establishes NP-completeness for recognizing leaf powers and pairwise compatibility graphs, extending the hardness to related graph classes and their generalizations.

Findings

Recognition problems are NP-complete.

Hardness extends to broader graph classes.

Implications for phylogenetic graph modeling.

Abstract

Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes remain poorly understood. In this paper, we establish that the recognition problem for both classes is NP-complete. We extend this hardness result to a broader hierarchy of graph classes, including pairwise compatibility graphs and their generalizations, multi interval pairwise compatibility graphs.