Structure and Recognition of 3,4-leaf Powers of Galled Phylogenetic Networks in Polynomial Time

TL;DR

This paper investigates the structure and recognition algorithms for 3,4-leaf powers of galled phylogenetic networks, extending traditional tree-based models to include certain cycles, with polynomial-time solutions for specific cases.

Contribution

It introduces structural insights and polynomial algorithms for recognizing 3 and 4-leaf powers of galled phylogenetic networks, expanding beyond tree models.

Findings

Polynomial algorithms for k=3 and k=4 leaf powers

Structural characterization of galled phylogenetic networks

Recognition of squares of galled networks in polynomial time

Abstract

A graph is a $k$-leaf power of a tree $T$ if its vertices are leaves of $T$ and two vertices are adjacent in $T$ if and only if their distance in $T$ is at most $k$. Then $T$ is a $k$-leaf root of $G$. This notion was introduced by Nishimura, Ragde, and Thilikos [2002] motivated by the search for underlying phylogenetic trees. We study here an extension of the $k$-leaf power graph recognition problem. This extension is motivated by a new biological question for the evaluation of the latteral gene transfer on a population of viruses. We allow the host graph to slightly differs from a tree and allow some cycles. In fact we study phylogenetic galled networks in which cycles are pairwise vertex disjoint. We show some structural results and propose polynomial algorithms for the cases $k=3$ and $k=4$. As a consequence, squares of galled networks can also be recognized in polynomial time.