On strongly chordal graphs that are not leaf powers

TL;DR

This paper investigates the relationship between strongly chordal graphs and leaf powers, providing a counterexample family and exploring the complexity of recognizing leaf powers, which are important in phylogenetics.

Contribution

It demonstrates that strongly chordal graphs are not always leaf powers by constructing an infinite family of counterexamples and analyzes the complexity of recognizing leaf powers.

Findings

Counterexamples show not all strongly chordal graphs are leaf powers

Deciding if a chordal graph is G-free is NP-complete

Connection established between leaf powers, alternating cycles, and quartet compatibility

Abstract

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem.