Nonbinary tree-based phylogenetic networks

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by  Laura Jetten

1
Nonbinary tree-based
phylogenetic networks
Laura Jetten and Leo van Iersel
Abstract
Rooted phylogenetic networks are used to describe evolutionary histories that contain non-treelike
evolutionaryeventssuchashybridizationandhorizontalgenetransfer.Insomecases, suchhistoriescan
be described by a phylogenetic base-tree with additional linking arcs, which can for example represent
gene transfer events. Such phylogenetic networks are called tree-based. Here, we consider two possible
generalizations of this concept to nonbinary networks, which we call tree-based and strictly-tree-based
6 nonbinaryphylogeneticnetworks.Wegivesimplegraph-theoreticcharacterizationsoftree-basedandstrictly-
1 tree-basednonbinaryphylogeneticnetworks.Moreover,weshowforeachofthesetwoclassesthatitcanbe
0 decidedinpolynomialtimewhetheragivennetworkiscontainedintheclass.Ourapproachalsoprovides
2 a new view on tree-based binary phylogenetic networks. Finally, we discuss two examples of nonbinary
p phylogeneticnetworksinbiologyandshowhowourresultscanbeappliedtothem.
e
S IndexTerms

0 Phylogenetictree,phylogeneticnetwork,evolution,tree-of-life,tree-based
3
✦

]
E
1 INTRODUCTION
P
.
o Rooted phylogenetic networks are becoming increasingly popular as a way to describe evolutionary
i historiesthatcannotbedescribedbyaphylogenetictree[1],[2].Theleavesofsuchanetworkarelabelledand
b
represent,forexample,currently-livingspecies,whiletherootofthenetworkrepresentsacommonancestor
-
q of those species. Vertices with two or more outgoing arcs represent a divergence event in which a lineage
[ splitinto twoor more lineages,while vertices withtwo or more incoming arcs representa convergence of

differentlineagesintoasinglelineage.Thelattereventsarecalledreticulateevolutionaryeventsandinclude,
3
v for example, hybridization, introgression and horizontal gene transfer. Therefore, these vertices are called
4 reticulations.Aphylogeneticnetworkwithoutreticulationsisa(rooted)phylogenetictree.Hence,phylogenetic
7 networksareamoregeneralmodelforevolutionaryhistoriesthanphylogenetictrees.
9
Although the occurence of reticulate evolutionary events is well-accepted, there are different views on
4
their importance. One possibility is to see evolution as a mainly tree-like (vertical) process with sporadic
0
horizontalevents.Theotherextremeistocompletelyabandontheideaofatree-of-lifeandtoseeevolution
.
1 purelyas a network [3], [4], [5], [6]. This discussionis especiallyrelevant for prokaryotes, where the main
0
formofnon-treelikeevolutionishorizontalgenetranser,i.e.geneticmaterialistransferredfromonespecies
6
toanothercoexistingspeciesthatisnotadescendant.Iftheevolutionaryhistoryofagroupofprokaryotes
1
: is mainly tree-like, then you could describe such a history as a phylogenetic species tree with additional
v
cross-connectingarcsdescribingthehorizontalgenetransferevents.However,iftheirevolutionisinherently
i
X network-like,thenitmightnotbepossibletoidentifyanytree-likesignalatall.
Thisdiscussionhasrecentlyledtotheintroductionofanewclassofphylogeneticnetworkscalled“tree-
r
a based”[7],whichcontainsthosenetworksthatcanbedescribedbyaphylogeneticbase-treewithadditional
linkingarcsbetweenbranchesofthebase-tree.Thisnotionwasmotivatedbytheobservationthatthisisnot
alwayspossible,i.e.thereexistnetworksthatcannotbedescribedasabase-treewithlinkingarcs[8].
Francis andSteelshowed recentlythat there is a polynomial-time algorithm todecide whethera given
binary phylogenetic network is tree-based or not [7]. In addition, it was shown that any phylogenetic
network can be made tree-basedby the addition of leaves. Hence, this notion has to be used with caution
in the presence of possible extinctions or under-sampling. Even more recently, a simple graph-theoretic
characterizationwasgiventhatcanalsobeusedtodecidewhetheragivenbinarynetworkistree-basedor
not [9]. Unfortunately, these results are all restrictedto binary phylogenetic networks, in which all vertices
• E-mail:[email protected]@gmail.com
• DelftUniversityofTechnology,PO-box5,2600AA,Delft,
TheNetherlands.
• LeovanIerselwaspartlysupportedbyaVidigrantfromtheNetherlandsOrganisationforScientificResearch(NWO)andpartlybythe
4TUAppliedMathematicsInstitute(4TU.AMI).
2
Reticulationnumber Leaves ReticulateProcess Tree-based Reference
Violets 21 16 polyploidisation no [15,Figure4],Figure19
OriginofEukaryotes 6 41 endosymbiosis yes [16,Figure2],Figure20
Influenza 5 7 reassortment yes [17,Figure1]
Cichlids(fish) 5 19 hybridisation yes [18,Figure4]
TABLE1
Examplesofnonbinaryphylogeneticnetworksinbiology.Thereticulationnumber isdefinedasthetotalnumberof“additional
branches”inthenetwork,i.e.areticulationwithpparentsaddsp−1tothereticulationnumber.
have at most two incoming and at most two outgoing arcs (see the next section for precise definitions).
Moreover,thetechniquesusedbytheseauthorsdonot(easily)extendtononbinarynetworks.
Here,wealsoconsidernonbinaryphylogeneticnetworks.Insuchanetwork,avertexcanhavemorethan
twooutgoingarcs, representinguncertaintyin the orderof divergence events,ormore thantwo incoming
arcs, representing uncertainty in the order of reticulate events. See Table 1 for real biological examples of
nonbinaryphylogeneticnetworks (alsosee[10], [11], [12]). Ingeneral, suchuncertaintiescannot simplybe
overcome by collecting more data [13], [14]. Since the tree-basedness of nonbinary networks has not been
introducedorstudiedbefore,wediscussdifferentpossibledefinitionsoftree-basedinthenonbinarycase.
Roughlyspeaking,wecallanonbinaryphylogeneticnetworkstrictlytree-basedifitcanbeobtainedfrom
a rooted (nonbinary) phylogenetic tree by adding linking arcs between the branches of the tree, such that
notwolinkingarcsattachatthesamepoint.Consequently,insuchanetworkallverticeshaveatmosttwo
incomingarcs.Inaddition,allverticeswithmorethantwooutgoingarcscorrespondtoverticesofthebase-
tree, becausethe newvertices that are created bythe addition of linking arcs all get at most two outgoing
arcs: one of the base-tree and one linking arc. Hence, a strictly-tree-based network can be nonbinary only
becausethebase-treecanbenonbinary.
We alsoconsider tree-basednonbinary phylogeneticnetworks, which arenetworks that canbe obtained
from a rooted (nonbinary) phylogenetic tree by adding linking arcs between branches and/or vertices of
thetree.Itturnsoutthatanetworkisinthisclasspreciselyifithasatleastone binaryrefinementthatisa
tree-basedbinaryphylogeneticnetwork.Thisisamoregeneralclassthanthestrictly-tree-basedvariant.
Our mainresultsareas follows. We first presentanalternativeviewon binarytree-basedphylogenetic
networks,whichcanpartlybeextendedtononbinarynetworks.Weintroduceomnians,whichwedefineas
non-leaf vertices of which all children are reticulations. We then use this notion to obtain a new, simpler
characterizationof binary tree-basedphylogenetic networks. We showthat a binary phylogeneticnetwork
is tree-based if and only if every subset S of its omnians has at least |S| different children. We use this to
derive,inanalternative(independentlydiscovered)way,thecharacterizationofbinarytree-basednetworks
in terms of zig-zag paths [9] and a new matching-based algorithm for deciding whether a given binary
networkistree-based.Wealsoshowthateverybinarynetworkwithatmosttworeticulationsistree-based
andgiveanewsufficientconditionforabinarynetworktobetree-based.
We then proceed to nonbinary networks. We show that our characterization of binary tree-based phy-
logenetic networks in terms of omnians can easily be generalized to the nonbinary case. We then obtain
the first polynomial-time algorithm for deciding whether a nonbinary phylogenetic network is tree-based.
Additionally,weshowasimplecounterexample,showingthatthecharacterizationbasedonzig-zagpaths
can not be used to characterize nonbinary tree-based networks. However, we also show that nonbinary
strictly-tree-based phylogenetic networks can be characterized using zig-zag paths. Consequently, also for
thisclassofnetworksitcanbedecidedinpolynomialtimewhetheragivennetworkbelongstotheclass.
We also discuss “stable” phylognetic networks [19], in which for each reticulation r there exists some
leaf x such that all paths from the root to x go through r. We show that, although all binary stable
phylogeneticnetworksaretree-based,thisisnotalwaysthecasefornonbinarynetworks.
This paper is organized as follows. We first give the definitions and new results for binary networks
in Section 2, then the definitions and results for nonbinary networks in Section 3. Examples of how these
results can be appliedto real, biological, nonbinary phylogenetic networks are given in Section 4. We end
withadiscussioninSection5.
2 BINARY PHYLOGENETIC NETWORKS
2.1 Preliminaries
First, some essential concepts around binary phylogenetic networks will be explained. Phylogenetic net-
workscontainverticesanddirectededges.Directededgeswillbecalledarcsfromnowon.
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Definition 1. A(rooted)binaryphylogeneticnetworkisadirectedacyclicgraphN =(V,A),whichcontainsa
singlerootwithindegree0andoutdegree1or2andmayinadditioncontainthefollowingtypesofvertices:
• verticeswithoutdegree0,calledleaves,whicharelabelled;
• verticeswithindegree2andoutdegree1,calledreticulations;
• verticeswithindegree1andoutdegree2,calledtree-vertices.
An example of a binary phylogenetic network is given in Figure 1, in which leaves are coloured blue
and reticulations are indicated with a pink shading around the nodes. A (rooted) binary phylogenetic tree is
a binary phylogenetic network that contains no reticulations. Although every arc is drawn without arrow
head, they are all directed to the lowest vertex. This is the case throughout the paper, unless explicitly
mentionedotherwise.
Root
y
x
c
a
b
Fig.1.Anexampleofabinaryphylogeneticnetworkwithleaflabelsa,bandc,whichcane.g.representthreepresent-dayspecies,
andomniansxandy.
Take(u,v)=a∈A,anarcfromvertexutov.Then,aiscalledanoutgoingarcofuandanincomingarc
ofv.Vertexuisaparentofvandviscalledachildofu.Ifthereisalsoanarc(u,w)∈A,thenvertexwandv
haveajointparent,sowandvarecalledsiblings.Whenanon-leafvertexzhasonlyreticulationsaschildren,
then z is called an omnian. For example in Figure 1, vertices x and y are omnians, since both children of
theseverticesarereticulations.Verticescanbeomnianandreticulationatthesametime,seee.g.verticesu
andv inFigure3.Becauseoftheimportanceofomnians,whichwillbecomeclearlateroninthepaper,we
alwaysusesquarenodesforomniansandcircularnodesforallothervertices.
Definition2. AbinaryphylogeneticnetworkN istree-basedwithbase-treeT,whenN canbeobtainedfromT via
thefollowingsteps:
(i) AddverticestothearcsofT.Thesevertices,calledattachmentpoints,havein-andoutdegree1.
(ii) Addarcs,calledlinkingarcs,betweenpairsofattachmentspoints,sothatN remainsbinaryandacyclic.
(iii) Suppresseveryattachmentpointthatisnotincidenttoalinkingarc.
Note in particular that it is not allowed to create multiple linking arcs between the same pair of
attachmentpointssinceN isrequiredtobeabinaryphylogeneticnetwork.
Abinaryphylogeneticnetworkistree-basedifitistree-basedwithbase-treeT forsomebinaryphyloge-
netictreeT.AnexampleoftheprocedureisdisplayedinFigure2.Anexampleofabinarynetworkthatis
nottree-basedisgiveninFigure3.
Arootedspanningtreeτ ofaphylogeneticnetworkN isasubgraphofN thatisarootedtreeandcontains
allvertices(andasubsetofthearcs)ofN.Adummyleaf ofarootedspanningtreeτ isavertexthatisnota
leafinnetworkN,butisaleafinτ.Hence,abinaryphylogeneticnetworkistree-basedifandonlyifithas
arootedspanningtreewithoutdummyleaves.
We will make heavy use of the following bipartite graph. Let N = (V,A) be a binary phylogenetic
network. The bipartite graph associated to N is the bipartite graph B = (U ∪ R,E) containing a vertex
vo ∈ U for each v ∈ V that is an omnian, a vertex wr ∈ R for each w ∈ V that is a reticulation, and an
edge {vo,wr} ∈ E for each(v,w) ∈ Awithv andomnianandw a reticulation.Hence,for avertex v ∈ V
thatisareticulationandanomnian,thereisavertexvoinU aswellasavertexvr inR.Foreaseofnotation,
wewillomitthesubscriptsfornowonandrefertovo andvr simplyasv.AnexampleisgiveninFigure4.
Let N = (V,A) be a binary phylogenetic network. An antichain is a set of vertices K ⊆ V for which
thereisnodirectedpathfrom anyvertexinK toanyothervertexinK.NetworkN satisfiestheantichain-
to-leafpropertyiffor everyantichaininN thereexistsapathfrom everyvertexinK toaleaf,sothatthese
pathsarearc-disjoint.Whichmeans,forexample,thatifthereisanantichainofthreeverticesandthereare
onlytwoleavesinthenetwork,thenetworkdoesclearlynotsatisfytheantichain-to-leafproperty.
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a b c a b c a b c
(a) (Base-)treeT (b) Attachmentpointsadded (c) Linkingarcsadded
c
a
a b c b
(d) Attachmentpointssuppressed (e) Redrawn as the network from
Figure1
Fig.2.Fromphylogenetictreetophyogeneticnetworkinsteps(a)to(e),whichshowsthatthenetworkfromFigure1istree-based.
α α
1 2
u
i
v
x j c
a
b
Fig.3.Anexampleofanon-tree-basedbinaryphylogeneticnetwork.Becausearcsiandjaretheonlyoutgoingarcsofverticesu
andvrespectively,theywouldbothhavetobepresentinthebase-tree.However,vertexxwouldthenhavetwoincomingarcsin
thebase-tree,whichisnotallowed.
An example of an antichain can be seen in Figure 3, where vertices α1 and α2 form an antichain. The
network does not satisfy the antichain-to-leaf property, because when we look at the antichain formed by
verticesuandv,therearenoarc-disjointpathstoleaves.
Avertexv iscalledstableifthereexistsaleaflforwhicheverypathfromtheroottolpassesthroughv.
Anetworkiscalledstableifeveryreticulationisstable.
Let G = (V,E) be a graph. If v,w ∈ V so that(v,w) ∈ E, then w is a neighbour of v. For a set S ⊆ V,
the neighbours of S are denoted by Γ(S). A matching M ⊆ E is a set of edges so that no vertex v ∈ V is
incidentwithmorethanoneedgeinM.AmaximalpathinGisapaththatisnotcontainedinalargerpath.
Thefollowingknownresultswillbeuseful.
Proposition 2.1. [7]ConsiderabinaryphylogeneticnetworkNoverleafsetX.
(i) IftheparentsofeachreticulationofN aretree-vertices,thenN istree-based.
(ii) IfN hasareticulationwhoseparentsarebothreticulations,thenN isnottree-based.
Proposition 2.2. [19]Inastable binaryphylogeneticnetworkN, thechildand theparentsofeach reticulationare
tree-vertices.
Thefollowingobservationfollowsdirectlyfromtheprevioustwopropositions.
Corollary2.3. Everybinarystablephylogeneticnetworkistree-based.
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2.2 Results
Thefollowingtheoremwillbeusedtoobtainasimplegraph-theoreticcharacterizationofbinarytree-based
phylogeneticnetworks.
Theorem 2.4. Let N be a binary phylogenetic network and B = (U ∪R,E) the bipartite graph associated to N.
NetworkN istree-basedifandonlyifthereexistsamatchingM inB with|M|=|U|.
Proof. AssumethereexistsamatchingM inBwith|M|=|U|,i.e.,allomniansarecoveredbyM.Construct
asetAofarcsasfollows:addtheoutgoingarcofeveryreticulationandtheincomingarcofalltree-vertices
toA.Additionally,foreachedgeofM,addthecorrespondingarcofN toA,ifithasnotyetbeenadded.For
everyreticulationthathasnotyetbeencovered,addoneofitsincommingarcstoA,arbitrarily.ThetreeT,
consistingofallverticesofN andthesetofarcsA,isarootedspanningtree,becausethereispreciselyone
incoming arc of every vertex contained in T. Moreover, there are no dummy leaves, because U is covered
byM.Hence,itfollowsthatN istree-based.
Now, assumethatN is tree-basedwithbase-treeT.Colour everyedge of B thatcorresponds to anarc
in T.When an omnianhas outdegree 2 and bothoutgoing arcs are contained in T, decolourize one of the
twocorrespondingedgesofB,arbitrarily.Hence,eachvertexofU isincidenttoatmostonecolourededge.
Since T is a rooted tree, it contains at most one incoming arc of each reticulation. Hence, also each vertex
ofRisincidenttoatmostonecolourededge.SothecolourededgesofB formamatchingM.BecauseT is
abase-tree,therearenodummyleaves,andsoallomniansarecoveredbyM.
This theorem can be used to verify whether a binary phylogenetic network N is tree-based or not in
polynomialtime,usinganalgorithmformaximumcardinalitybipartitematching(seee.g.[20]).
WewilllookatanexampleofabinaryphylogeneticnetworkN andthebipartitegraphB =(U ∪R,E)
associatedtoN inFigure4.Sincethereexistsamatching,whichiscolouredblueanddash-dottedinFigure
4(b),thatcoversU,thebinaryphylogeneticnetworkinFigure4(a)istree-based.Abase-treeT ofnetworkN
can be seen in Figure 5, where the arcs that correspond to edges of the matching are dash-dotted and
linking-arcsaredashed.
U R
y y
v
z
y z z
q x x
a d
b c v q
(a) A rooted binary phylogenetic net- (b) The bipartite graph asso-
workN. ciated to N, with a match-
ing covering U indicated by
dash-dottedlines.
Fig.4.UsingTheorem2.4toshowthatthisisatree-basedbinaryphylogeneticnetwork.
v
y z
q x
a d
b c a b c d
Fig.5.Abase-treeT ofthenetworkinFigure4(a).
Since a binary phylogenetic network that contains no reticulations is a rooted tree, such a network
is clearly tree-based. The next theorem shows that this is still the case for all networks with one or two
reticulations.Ontheotherhand,Figure6showsapartofanetworkN thatcontainsthreereticulationsand
isnottree-based.Soitfollowsthatnotallnetworkswiththreereticulationsaretree-based.
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Theorem2.5. IfabinaryphylogeneticnetworkN containsatmosttworeticulations,thenN istree-based.
Proof. If N contains only one reticulation, then both parents of this reticulation are tree-vertices and with
Proposition2.1itfollowsthatN istree-based.
NowconsiderthecasethatN containsexactlytworeticulationsxandy.Ifxandydonothaveaparent-
child relation, then both parents of x and y are tree-vertices and it follows from Proposition 2.1 that N is
tree-based.Nowsupposethatxistheparentofy.Therearetwopossibilities,xandy havingajointparent
andxandy havingdifferentparents,bothdisplayedinFigure7.
Fig.6.Localsituationinanetworkthathasthreereticulationsandisnottree-based.
z
z
w
x x
y y
N N
1 2
(a) (b)
Fig.7.Thetwopossibilitiesthatcanoccurwhenreticulationxistheparentofreticulationy,usedintheproofofTheorem2.5.
FrompartialnetworksN1andN2ofFigure7wecreatetwobipartitegraphs,A=(U ∪R,E)associated
toN1 andB =(U ∪R,E)associatedtoN2,thataredisplayedinFigure8.
U R U R
z x x
x
x y y
(a) BipartitegraphAassoci- (b) BipartitegraphBassoci-
atedtonetworkN1. atedtonetworkN2.
Fig.8.ThebipartitegraphsassociatedtothepartialnetworksinFigure7.
In both cases in Figure 8 it is easy to see that there is a matching that covers U. It then follows from
Theorem2.4thatN istree-based.
Toobtainasimplecharacterizationofbinarytree-basednetworks,wewilluseHall’sTheorem,whichis
statedbelow.
Theorem2.6(Hall’sTheorem[21]). LetB =(U ∪W,E)beabipartitegraph.ThereexistsamatchinginB that
coversU ifandonlyif,foreveryU1 ⊆ U,thenumberofdifferentneighboursoftheverticesinU1 isatleast|U1|.
Consider Hall’sTheorem andTheorem 2.4. Combiningthose twotheorems givesa characterizationfor
abinaryphylogeneticnetworktobetree-based.
Corollary 2.7. Let N be a binaryphylogeneticnetwork and U the set of all omnians of N. Then N is tree-basedif
andonlyifforallS ⊆U thenumberofdifferentchildrenoftheverticesinS isgreaterthanorequaltothenumberof
omniansinS.
7
Proof. FollowsdirectlyfromTheorem2.4andTheorem2.6.
AnexampleofhowthistheoremandcorollarycanbeappliedisgiveninFigure9,whereanexampleof
abinaryphylogeneticnetworkN isdisplayedin(a)andthebipartitegraphB =(U ∪ R,E)associatedto
N in(b).Omniansareindicatedassquarenodesandreticulationsaremarkedwithapinkshadingaround
thenodes.
From the bipartite graph in Figure 9 it follows with Hall’s Theorem, with S = U, that there exists no
matching in B that covers U. Therefore, with Theorem 2.4 it follows that N in Figure 9(a) is not tree-
based. Indeed, we can directly see in N that the omians in S = {a,i,h,f,g} have only four different
children{b,c,d,e}(notethat,ingeneral,thesetS andthesetofchildrenofverticesinS donothavetobe
disjoint).Hencethisnetworkisnottree-based.
a
i
b
U R
p
h f a
x
b
a
c
q c
f
g i
y d
h
e
e d
w d g f
Fig.9.Exampleofanon-tree-basedbinaryphylogeneticnetworkN andthebipartitegraphBassociatedtoN.
Inapreviousarticle,thefollowingnecessaryconditionforanetworktobetree-basedwasfound.
Proposition 2.8. [7]IfabinaryphylogeneticnetworkoverleafsetX istree-based,thenitsatisfiestheantichain-to-
leafproperty.
On the other hand, if a network is not tree-based, it can still satisfy the antichain-to-leaf property, as
shownbytheexamplefrom[7]inFigure10.However,itisdifficulttoseeinthisnetworkwhatisgoingon.
Therefore,weshowinFigure11twoexamplesoflocalstructuresthatcauseanetworktobenon-tree-based.
The example inFigure 11(a) is similar to the local structure that causesthe network of Figure 10 not tobe
tree-based,whichcanbecheckedusingCorollary2.7.
At first sight,it might look like networks containingthe localstructures inFigure 11cannot satisfythe
antichain-to-leafproperty.However,inFigure12weshowthatitisindeedpossiblethattheydosatisfythis
property. Although it was already known that networks that satisfy the antichain-to-leaf property are not
necessarilytree-based,thesefiguresilluminatewhythisispossible.
Looking at the examples in Figure 11, we see that a pattern has emerged. In (a) the pattern is marked
dashed in green. Starting at vertex o1 and ending at vertex o5, we see a zigzag starting with an omnian,
alternatingbetweenreticulations and omnians, eventuallyending withan omnian. The last omnianin the
patterncanbeareticulationthatisalreadypartofthepath,ascanbeseeninFigure11(b).
The next theorem shows that every binaryphylogenetic network that is not tree-basedcontains a local
structureasintheexamplesinFigure11.
Theorem 2.9. Let N be a binary phylogenetic network and B = (U ∪R,E) the bipartite graph associated to N.
NetworkN istree-basedifandonlyifB containsnomaximalpathwhichstartsandendsinU.
Proof. Notice that every vertex in B is of degree at most 2. Therefore, B is a disjoint union of paths and
cycles.Hence,foreachconnectedcomponentB′ =(U′∪R′,E′)ofB,therearefourpossibletopologies:
8
o
2
o
1
r
1
o
3
r
2
Fig.10.Nottree-basedbinaryphylogeneticnetworksatisfyingtheantichain-to-leafproperty[7].
o o
o 2 3
1
r1 =o4 r2
o o o o o
1 2 3 4 5
r r r r
1 2 3 4
r
3
(a) (b)
Fig.11.Examplesoflocalstructuresofbinaryphylogeneticnetworksthatarenottree-based.
i) AmaximalpaththatbeginsandendsinR.
ii) AmaximalpaththatbeginsinU andendsinR.
iii) AmaximalpaththatbeginsandendsinU.
iv) Acircuit.
i)AllverticesinRareofdegreeatmost2.BecausethemaximalpathbeginsandendsinR,allvertices
inU′havedegree2.LetS ⊆U′.RecallthatΓ(S)denotesthesetofneighboursofverticesinS.Thenumber
of edges incident toS = 2|S| ≤ the numberof edges incident to Γ(S) ≤ 2|Γ(S)|. Therefore,|S| ≤ |Γ(S)|
forallS ⊆U′.ItfollowsfromHall’sTheoremthatthereexistsamatchinginB′ thatcoversU′.
ii) All vertices in R are of degree at most 2. All vertices in U′ have degree 2, except for the omnian o1
wherethemaximalpathbegins.LetS ⊆U′.ConsiderthesubgraphofB′inducedbyS∪Γ(S).Itconsistsof
paths.Firstconsidersuchapaththatdoesnotcontaino1.ThenthepathmustbeginandendinR,because
every omnian in S except for o1 has two neighbours in Γ(S). Hence, the paths contain more reticulations
than omnians. Now consider a path that contains o1. Then it is a path that begins in U and ends in R.
Hence,itcontainsasmanyomniansasreticulations.Itfollowsthatallpathstogether,i.e.thesubgraphofB′
inducedbyS∪Γ(S),containsmorereticulationsthanomnians,so|S|≤|Γ(S)|.Again,sincethisholdsfor
allS ⊆U′.ItfollowsfromHall’sTheoremthatthereexistsamatchinginB′ thatcoversU′.
iii) LetS = U′. Then|S| = |Γ(S)|+1. Hence,itfollows from Hall’sTheoremthattheredoes not exista
matchinginB thatcoversU.
iv) All vertices in B are of degree 2 and it follows in the same way as in case i) that |S| ≤ |Γ(S)| for
allS ⊆U′ andhencethatthereexistsamatchinginB′ thatcoversU′.
9
(a) (b)
Fig.12.Localstructuresofbinaryphylogeneticnetworksthatsatisfytheantichain-to-leafpropertybutarenottree-based.
Hence, there exists a matching in B that covers U precisely if there is no maximalpath that starts and
endsinU.ThetheoremnowfollowsfromTheorem2.4.
Proposition 2.1 showed that a binary phylogenetic network is tree-based if for each reticulation both
parents are tree-vertices and not tree-based if for at least one reticulation both parents are reticulations.
However, inthe situationinwhich a reticulationin N has one parentthat is a reticulationand the other a
tree-vertexitisnotimmediatelyclearifN istree-basedornot.Thenextcorollaryshowsthatsuchnetworks
aretree-basedifanadditionalconditionisfulfilled.
Corollary2.10. IfforeveryreticulationrinabinaryphylogeneticnetworkN either
(i) bothparentsofraretree-vertices;or
(ii) oneparentofrisatree-vertexandthesiblingofrisatree-vertexoraleaf,
thenN istree-based.
Proof. LetB =(U∪R,E)bethebipartitegraphassociatedtoN.SinceallverticesinBhavedegreeatmost
two, each connected component of B is a pathor a cycle. Assumethat there exists a maximalpathP that
startsandendsinU.Letu∈U bethefirstvertexonthispath.Sinceuisanomnianwithonlyonechild,itis
alsoareticulation.Letrbetheonlychildofu.Sinceoneparentofrisareticulation,theotherparentpofr
mustbe a tree-vertex andthe siblingof r alsoa tree-vertexor a leaf. However, that meansthat p is not an
omnian.Hence,pathP endsinr, whichisacontradictiontotheassumptionthatP endedinU.Itfollows
thatthereisnopaththatstartsandendsinU.ByTheorem2.4itfollowsthatN istree-based.
The following characterization of binary tree-based phylogenetic networks follows directly from The-
orem 2.9, thus providing an alternative proof of this characterization which was independently discov-
ered (in a slightly different form) by Louxin Zhang [9]. See Figure 11 for examples. We call a sequence
(u1,v1,...,uk,vk,uk+1) of 2k +1 vertices (k ≥ 1) of a network N a zig-zag path if vi is the child of ui
andui+1 fori=1,...,k.
Corollary 2.11. A binary phylogeneticnetwork is tree-based if and only if it contains no zig-zag path (o1,r1,...,
ok,rk,ok+1), with k ≥ 1, in which r1,...,rk are reticulations, o1,...,ok+1 are omnians and o1 and ok+1 are
reticulationsaswellasomninans.
3 NONBINARY PHYLOGENETIC NETWORKS
3.1 Preliminaries
Westartwiththedefinitionofnonbinarynetworks.1 AnexampleisgiveninFigure13.
1.Wheneverwerefertononbinary,wemean“not-necessarily-binary”.
10
Definition 3. A(rooted)nonbinaryphylogeneticnetworkisadirected,acyclicgraphN =(V,A)thatcontains
asinglerootwithindegree0andoutdegree1ormoreandmayadditionallycontainthefollowingtypesofvertices:
• verticeswithindegree1andoutdegree0,calledleaves(colouredblueinFigure13),whicharelabelled;
• verticeswithoutdegree1andindegree2ormore,calledreticulations(markedinpinkinFigure13);
• verticeswithindegree1andoutdegree2ormore,calledtree-vertices.
Root
a b d e f g
c
Fig.13.Exampleofanonbinaryphylogeneticnetwork.
Anonbinaryphylogenetictreeisanonbinaryphylogeneticnetworkwithoutreticulations.
Wewillconsidertwodifferentvariantsoftree-basednessofnonbinarynetworks,whichwename“tree-
based”and“strictly-tree-based”.
Definition 4. AnonbinaryphylogeneticnetworkN iscalledtree-basedwithbase-treeT,whenN canbeobtained
fromT viathefollowingsteps:
(a) AddsomeverticestoarcsofT.Thesevertices,calledattachmentpoints,havein-andoutdegree1.
(b) Addarcs,calledlinkingarcs,betweenpairsofattachmentspointsandfromtree-verticestoattachmentpoints,
sothatN remainsacyclicandsothatattachmentpointshaveindegreeoroutdegree1.
(c) Suppresseveryattachmentpointthatisnotincidenttoalinkingarc.
Definition 5. A nonbinary phylogeneticnetwork N is called strictly tree-basedwith base-tree T,whenN can be
obtainedfromT viathefollowingsteps:
(1) AddsomeverticestoarcsinT.Thesevertices,calledattachmentpoints,havein-andoutdegree1.
(2) Add arcs, called linking arcs, between pairs of attachments points, so that N remains acyclic and so that
exactlyonelinkingarcisattachedtoeachattachmentpoint.
A nonbinary phylogenetic network is tree-based if it is tree-based with base-tree T for some nonbinary
phylogenetictreeT.Similarly,anonbinaryphylogeneticnetworkisstrictlytree-basedifitisstrictlytree-based
withbase-treeT forsomenonbinaryrootedphylogenetictreeT.
The distinction between tree-based and strictly tree-based is illustrated by two examples in Figure 14.
Anexampleofastrictly-tree-basednonbinarynetworkcanbefoundinFigure18(a).
a b c d e a b c d
(a) Thisnetworkisnotstrictly (b)Thisnetworkisnotstrictly
tree-basedbecausethelinking tree-based because two link-
arc is attached to a vertex of ing arcs are attached to the
the base-tree, rather than to same attachment point, and
an attachmentpoint, and this thiscannotbeavoided.
cannotbeavoided.
Fig. 14. Two nonbinary networks that are both tree-based but not strictly tree-based. The black solid lines indicate possible
base-trees,whilethedashed,greylinesarethelinkingarcs.
Thefollowingobservationsareeasilyverified.