On Theexistenceofinfinitely

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Letter to Editor

On the existence of in nitely many universal tree-based networks

a r t i c l e i n f o

Keywords:Reticulate evolutionbinary phylogenetic networktree-based network

2010 MSC:

05C05 (Primary)05C2005C25

a b s t r a c t

A tree-based network on a set X of n leaves is said to be universal if any rooted binary phylogenetic treeon X can be its base tree. Francis and Steel showed that there is a universal tree-based network on X inthe case of n ¼ 3, and asked whether such a network exists in general. We settle this problem by provingthat there are in nitely many universal tree-based networks for any n 4 1.

& 2016 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-NDlicense ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

1. Introduction

Throughout this paper, n denotes a natural number that isgreater than 1 and X represents the set f1 , 2 , …, ng. All graphsconsidered here are directed acyclic graphs. A graph G0 is said to bea subdivision of a graph G if G0 can be obtained from G by insertingvertices into arcs of G zero or more times. Given a vertex v of agraph with indeg ðvÞ ¼outdeg ðvÞ ¼1, smoothing (or suppressing ) vrefers to removing v and then adding an arc from the parent to thechild of v . Two graphs are said to be homeomorphic if they become

isomorphic after smoothing all vertices of in-degree one and out-degree one.

For the reader's convenience, we brie y recall the relevantbackground from Francis and Steel (2015) (see Semple and Steel,2003 for the terminology in phylogenetics).

De nition 1.1. A rooted binary phylogenetic network on X isde ned to be a directed acyclic graph ( V , A) with the followingproperties:

X ¼ f v A V j indeg ðvÞ ¼1 , outdeg ðvÞ ¼0g; there is a unique vertex ρ A V with indeg ð ρ Þ ¼0 andoutdeg ð ρ ÞA f1 , 2g;

for all v A V f X [ f ρ gg, f indeg ðvÞ, outdeg ðvÞg ¼ f1 , 2g.

The vertices in X are called leaves , and the vertex ρ is called the root .

De nition 1.2. Suppose T ¼ ðV , AÞ is a rooted binary phylogenetictree on X . A rooted binary phylogenetic network N on X is said tobe a tree-based network on X with base tree T if there is a sub-division T 0 ¼ ðV 0, A0Þ of T and a set I of mutually vertex-disjointarcs between vertices in V 0 V such that ðV 0, A0[ I Þ is acyclic and ishomeomorphic to N . The vertices in V 0 V are called attachment points , and the arcs in I are called linking arcs .

Tree-based networks can have an important role to play inmodern phylogenetic inference as they can represent more

intricate or realistic relationships among taxa than phylogenetictrees without compromising the concept of ‘underlying trees ’ (cf .,Francis and Steel, 2015; Huson et al., 2010 ).

In order to state the problem formally, we now introduce thenotion of universal tree-based networks. A tree-based network on X is said to be universal if any binary phylogenetic tree on X can bea base tree. We can de ne universal tree-based networks in amore concrete manner with the number ð2n 3Þ!! of binary phy-logenetic trees on X as follows.

De nition 1.3. A tree-based network N

¼ ðV , AÞ on X is said to beuniversal if for any binary phylogenetic tree T ðiÞ on X ði A f1 , 2 , …, ð2n 3Þ!! gÞ, there is a set I ðiÞ A of linking arcs suchthat ðV , A I ðiÞÞ is homeomorphic to T ðiÞ.

Problem 1.4 (Francis and Steel, 2015 ). Does a universal tree-basednetwork on a set X of n leaves exist for all n?

This problem is fundamental because it explores whether aphylogenetic tree on X is always reconstructable from a tree-basednetwork on X . Francis and Steel (2015) pointed out that the answeris ‘yes ’ for n ¼ 3. In this paper, we will completely settle theirquestion in the af rmative and provide further insights into uni-versal tree-based networks ( Theorem 3.1 ).

2. Preliminaries

Here, we slightly generalise the concept of tree shapes. Given atree-based network N on X , ignoring the labels on the leaves of N results in an unlabelled tree-based network N with n leaves. Weuse the two different types of symbols, such as N and N , to meanunlabelled and labelled tree-based networks, respectively. Twotree-based networks N and N 0 on X are said to be shape equivalent if N and N 0 are isomorphic. This equivalence relation partitions aset of the tree-based networks on X into equivalence classes calledtree-based network shapes with n leaves .

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2016.02.0230022-5193/ & 2016 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Journal of Theoretical Biology 396 (2016) 204 – 206

http://www.sciencedirect.com/science/journal/00225193http://www.elsevier.com/locate/yjtbihttp://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://crossmark.crossref.org/dialog/?doi=10.1016/j.jtbi.2016.02.023&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jtbi.2016.02.023&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.jtbi.2016.02.023&domain=pdfhttp://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://dx.doi.org/10.1016/j.jtbi.2016.02.023http://www.elsevier.com/locate/yjtbihttp://www.sciencedirect.com/science/journal/00225193


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Acknowledgements

The author thanks Kenji Fukumizu and the anonymousreviewers for their careful reading and helpful comments.

References

Francis, A.R., Steel, M., 2015. Which phylogenetic networks are merely trees withadditional arcs? Syst. Biol. 64 (5), 768 – 777, http://dx.doi.org/10.1093/sysbio/syv037 .

Harding, E.F., 1971. The probabilities of rooted tree-shapes generated by random bifur-cation. Adv. Appl. Probab. 3 (1), 44 – 77, http://www.jstor.org/stable/1426329 .

Huson, D.H., Rupp, R., Scornavacca, C., 2010. Phylogenetic Networks: Concepts,Algorithms and Applications. Cambridge University Press, New York, NY, USA,ISBN-10: 0521755964, ISBN-13; 9780521755962 .

Semple, C., Steel, M., 2003. Phylogenetics, Oxford Lecture Series in Mathematics andits Applications, vol. 24, Oxford University Press, Oxford.

Zhang, L., 2015. On tree based phylogenetic networks, http://arxiv.org/abs/1509.01663 .

Momoko Hayamizu n

Department of Statistical Science, School of MultidisciplinarySciences, SOKENDAI (The Graduate University for Advanced Studies),

Tokyo 190-8562, JapanThe Institute of Statistical Mathematics, Tokyo 190-8562, JapanE-mail address: [email protected]

Available online 24 February 2016

n Correspondence address: The Institute of Statistical Mathematics, Tokyo 190-8562,

Japan.

Letter to Editor / Journal of Theoretical Biology 396 (2016) 204 – 206 206
http://dx.doi.org/10.1093/sysbio/syv037http://www.jstor.org/stable/1426329http://www.jstor.org/stable/1426329http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://arxiv.org/abs/1509.01663http://arxiv.org/abs/1509.01663http://arxiv.org/abs/1509.01663mailto:[email protected]:[email protected]://arxiv.org/abs/1509.01663http://arxiv.org/abs/1509.01663http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://refhub.elsevier.com/S0022-5193(16)00119-3/sbref3http://www.jstor.org/stable/1426329http://dx.doi.org/10.1093/sysbio/syv037http://dx.doi.org/10.1093/sysbio/syv037http://dx.doi.org/10.1093/sysbio/syv037

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