Andreas D. M. Gunawan

Locating a Tree in a Phylogenetic Network in Quadratic Time

A fundamental problem in the study of phylogenetic networks is to determine whether or not a given phylogenetic network contains a given phylogenetic tree. We develop a quadratic-time algorithm for this problem for binary nearly-stable phylogenetic networks. We also show that the number of reticulations in a reticulation visible or nearly stable phylogenetic network is bounded from above by a function linear in the number of taxa.

A program for verification of phylogenetic network models

MOTIVATION Genetic material is transferred in a non-reproductive manner across species more frequently than commonly thought, particularly in the bacteria kingdom. On one hand, extant genomes are thus more properly considered as a fusion product of both reproductive and non-reproductive genetic transfers. This has motivated researchers to adopt phylogenetic networks to study genome evolution. On the other hand, a genes evolution is usually tree-like and has been studied for over half a century. Accordingly, the relationships between phylogenetic trees and networks are the basis for the reconstruction and verification of phylogenetic networks. One important problem in verifying a network model is determining whether or not certain existing phylogenetic trees are displayed in a phylogenetic network. This problem is formally called the tree containment problem. It is NP-complete even for binary phylogenetic networks. RESULTS We design an exponential time but efficient method for determining whether or not a phylogenetic tree is displayed in an arbitrary phylogenetic network. It is developed on the basis of the so-called reticulation-visible property of phylogenetic networks. AVAILABILITY AND IMPLEMENTATION A C-program is available for download on http://www.math.nus.edu.sg/∼matzlx/tcp_package CONTACT matzlx@nus.edu.sg SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

A decomposition theorem and two algorithms for reticulation-visible networks

In studies of molecular evolution, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree (resp. cluster) is contained in a phylogenetic network on the same taxa. These tree and cluster containment problems are known to be NP-complete. A phylogenetic network is reticulation-visible if every reticulation node separates the root of the network from at least a leaf. We answer an open problem by proving that the tree containment problem is solvable in quadratic time for reticulation-visible networks. The key tool used in our answer is a powerful decomposition theorem. It also allows us to design a linear-time algorithm for the cluster containment problem for networks of this type and to prove that every galled network with n leaves has 2 ( n - 1 ) reticulation nodes at most.

Solving the Tree Containment Problem for Genetically Stable Networks in Quadratic Time

A phylogenetic network is a rooted acyclic digraph whose leaves are labeled with a set of taxa. The tree containment problem is a fundamental problem arising from model validation in the study of phylogenetic networks. It asks to determine whether or not a given network displays a given phylogenetic tree over the same leaf set. It is known to be NP-complete in general. Whether or not it remains NP-complete for stable networks is an open problem. We make progress towards answering that question by presenting a quadratic time algorithm to solve the tree containment problem for a new class of networks that we call genetically stable networks, which include tree-child networks and comprise a subclass of stable networks.

Solving the tree containment problem in linear time for nearly stable phylogenetic networks

Abstract A phylogenetic network is a rooted acyclic digraph whose leaves are uniquely labeled with a set of taxa. The tree containment problem asks whether or not a phylogenetic network displays a phylogenetic tree over the same set of labeled leaves. It is a fundamental problem arising from validation of phylogenetic network models. The tree containment problem is NP -complete in general. To identify network classes on which the problem is polynomial time solvable, we introduce two classes of networks by generalizations of tree-child networks through vertex stability, namely nearly stable networks and genetically stable networks. Here, we study the combinatorial properties of these two classes of phylogenetic networks. We also develop a linear-time algorithm for solving the tree containment problem on binary nearly stable networks.

Solving the Tree Containment Problem for Reticulation-Visible Networks in Linear Time

The tree containment problem (TCP) is a fundamental problem in phylogenetic study. It was introduced as a mean for verifying whether a network is consistent with a binary tree. The containment problem is NP-complete, even if the network input is binary. If the input is restricted to reticulation-visible networks, the TCP has been proved to be solvable in quadratic time. In this paper, we show that there is a linear time TCP algorithm for binary reticulation-visible networks.

S-Cluster++: a fast program for solving the cluster containment problem for phylogenetic networks

Motivation Comparative genomic studies indicate that extant genomes are more properly considered to be a fusion product of random mutations over generations (vertical evolution) and genomic material transfers between individuals of different lineages (reticulate transfer). This has motivated biologists to use phylogenetic networks and other general models to study genome evolution. Two fundamental algorithmic problems arising from verification of phylogenetic networks and from computing Robinson‐Foulds distance in the space of phylogenetic networks are the tree and cluster containment problems. The former asks how to decide whether or not a phylogenetic tree is displayed in a phylogenetic network. The latter is to decide whether a subset of taxa appears as a cluster in some tree displayed in a phylogenetic network. The cluster containment problem (CCP) is also closely related to testing the infinite site model on a recombination network. Both the tree containment and CCP are NP‐complete. Although the CCP was introduced a decade ago, there has been little progress in developing fast algorithms for it on arbitrary phylogenetic networks. Results In this work, we present a fast computer program for the CCP. This program is developed on the basis of a linear‐time transformation from the small version of the CCP to the SAT problem. Availability and implementation The program package is available for download on http://www.math.nus.edu.sg/˜matzlx/ccp.

The Nk-valued Roman Domination and Its Boundaries

Abstract The Roman dominating function on a graph G = ( V , E ) is a labeling f : V → { 0 , 1 , 2 } satisfying that any vertex v with f ( v ) = 0 is adjacent to a vertex u with f ( u ) = 2 . In this paper, we generalize the notion of independence and dominance between two vertices. This gives a new generalization of Roman domination, called N k -valued Roman domination, where the codomain of the Roman dominating function is extended to N k = { 0 , 1 , 2 , … , k } . Two lower bounds of this N k -valued Roman domination number in terms of the diameter and radius of G respectively are established.