Recognizing and realizing cactus metrics

(1)

Delft University of Technology

Recognizing and realizing cactus metrics

Hayamizu, Momoko; Huber, Katharina T.; Moulton, Vincent; Murakami, Yukihiro

DOI

10.1016/j.ipl.2020.105916

Publication date

2020

Document Version

Final published version

Published in

Information Processing Letters

Citation (APA)

Hayamizu, M., Huber, K. T., Moulton, V., & Murakami, Y. (2020). Recognizing and realizing cactus metrics.

Information Processing Letters, 157, [105916]. https://doi.org/10.1016/j.ipl.2020.105916

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

(2)

Contents lists available atScienceDirect

Information

Processing

Letters

www.elsevier.com/locate/ipl

Recognizing

and

realizing

cactus

metrics

Momoko Hayamizu

a

,

b

,

Katharina

T. Huber

c

,

Vincent Moulton

c

,

∗

,

Yukihiro Murakami

d

a_The_Institute_of_Statistical_Mathematics,_190–8562,_Tachikawa,_Tokyo,_Japan b_JST_PRESTO,_190–8562,_Tachikawa,_Tokyo,_Japan

c_School_of_Computing_Sciences,_University_of_East_Anglia,_NR4_7TJ,_Norwich,_United_Kingdom

d_Delft_Institute_of_Applied_Mathematics,_Delft_University_of_Technology,_Van_Mourik_Broekmanweg_6,₂₆₂₈_XE,_Delft,_the_Netherlands

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received4August2019 Accepted13January2020 Availableonline16January2020 CommunicatedbyŁukaszKowalik

Keywords: Cactusmetric Metricrealization Optimalrealization Algorithms Phylogeneticnetwork

The problem of realizing ﬁnite metric spaces in terms of weighted graphs has many applications.Forexample,themathematicalandcomputationalpropertiesofmetricsthat canberealizedbytreeshavebeenwell-studiedandsuchresearchhaslaidthefoundation of the reconstruction of phylogenetic trees from evolutionary distances. However, as trees maybe toorestrictivetoaccuratelyrepresent real-worlddata orphenomena, itis importanttounderstandtherelationshipbetweenmoregeneralgraphsanddistances.In thispaper,weintroduceanewtypeofmetriccalledacactusmetric,thatis,ametricthat canberealizedbyacactusgraph.Weshowthat,justaswithtreemetrics,acactusmetric hasauniqueoptimalrealization.Inaddition,wedescribeanalgorithmthatcanrecognize whetherornotametricisacactusmetricand,ifso,computeitsoptimalrealizationin

O(n3)time,wheren isthenumberofpointsinthespace.

1. Introduction

Themetricrealizationproblem,whichistheproblemof representingafinitemetricspacebyaweightedgraph,has many applications, mostnotably inthe reconstruction of evolutionarytrees.Althoughanyfinitemetricspacecanbe realizedby a weighted complete graph,there canbe dif-ferentgraphs thatinduce thesamemetric.In[8], Hakimi and Yau first considered “optimal” realizations of finite metricspaces,whicharerealizationsofleasttotalweight. Althougheveryfinitemetricspacehasan optimal realiza-tion[6,12],theproblemoffindinganoptimalrealizationis NP-hard ingeneral[1,17] and the optimalsolutionisnot necessarilyunique[1,6].

*

Correspondingauthor.

E-mailaddresses:hayamizu@ism.ac.jp(M. Hayamizu),

k.huber@uea.ac.uk(K.T. Huber),v.moulton@uea.ac.uk(V. Moulton),

y.murakami@tudelft.nl(Y. Murakami).

A well-known special case of optimal realizations is provided by tree metrics, namely, those metrics that can berealizedbysomeedge-weightedtree.Foranytree met-ric on a ﬁnite set X , its optimal realizationis an X -tree (i.e.,atree inwhichsome verticesare labeledby X )and isuniquelydetermined[8].Inaddition,thereexistoptimal polynomial-time algorithms for computing the tree real-ization from a tree metric [3–5]. However, not much is knownaboutthepropertiesofoptimalrealizationsof met-ricsinduced by graphs that are more generalthan trees. Developing our understanding in this direction could be useful,astreescansometimesbetoorestrictivefor realiz-ingmetricsarisinginreal-worldapplications[11].

Inthispaper,wegeneralizetheconceptofatree met-ric by introducing a new type of metric called a “cac-tus metric1_” _which _can _be _realized _by _an _{edge-weighted} “ X -cactus”,whereacactusisaconnectedgraphinwhich

1 _This_concept_was_ﬁrst_introduced_in_[₉_]. https://doi.org/10.1016/j.ipl.2020.105916

(3)

2 M. Hayamizu et al. / Information Processing Letters 157 (2020) 105916

Fig. 1. An example of an X -cactus with a label-set X= {x1,. . . ,x16},

wheretheweightofeachedgeisproportionaltoitslength.Thevertices labeledbyanelementofX areshowninblack.Thewhitecirclesare ver-ticesthatarenotinX .

each edge belongs to at most one cycle. An example of an X -cactus is presented in Fig. 1. Note that cacti have some nicepropertiesincommonwithtrees.Forinstance, every cactus is planar andthe numberof vertices in an X -cactusisO

(

|

X

|)

aswith X -trees,whichmeansthat cac-tusmetricsareeasytovisualize.Inparticular,theyprovide a special case of an open problem in discrete geometry fromMatoušek[13].Besidestheseobservations,inthis pa-per we prove that, justas with tree metrics, any cactus metric hasaunique optimalrealization.We alsodescribe a polynomial time algorithm fordecidingwhether ornot anarbitrarymetricisacactusmetric,whichalsocomputes itsoptimalrealizationincaseitis.

2. Preliminaries

A metric onaset S isdeﬁnedtobe afunctiond

:

S

×

S

→ R

_≥0 withthepropertythatd equalszeroifandonly

if thetwo elements in S are identical,issymmetric, and satisﬁesthetriangleinequality.

All graphs considered here are ﬁnite,connected, sim-ple, undirected graphs in which the edges have positive weights. For any graph G, V

(

G

)

and E

(

G

)

represent the vertex-set andedge-set of G,respectively. Forany vertex v ofa graphG,thenumberofedges ofG thathavev as an endvertexis denoted bydeg

(

v

)

. Foranygraph G and any subset S of V

(

G

)

,we let dG denote themetric on S induced by takingshortestpaths in G between elements in S.

Throughoutthispaper,weusethesymbol X to repre-sentaﬁnitesetwith

|

X

|

≥

2,whichissometimescalleda label-set. Forany metricd on X , a realization of

(

X

,

d

)

is a graph G suchthat X isa subset ofV

(

G

)

andd

(

x

,

y

)

=

dG

(

x

,

y

)

holdsforeachx

,

y

∈

X ,whereweshallalways as-sumethateach vertexv ofG withdeg

(

v

)

≤

2 hasa label in X [12].Arealizationisminimal iftheremovalofan ar-bitrary edgeofG yieldsagraphthatdoesnotrealized.It isoptimal ifthesumofitsedgeweightsisminimumover all possiblerealizations(notethat optimalrealizationsare minimal buttheconversedoesnothold). Anyﬁnite met-ricspacehasatleastoneoptimalrealization[12,Theorem 2.2].

We now state a theorem concerning optimal realiza-tions which will be usefulin our proofs. For a graph G, each maximalbiconnectedsubgraphofG iscalleda block ofG andeachvertexofG sharedbytwoormoreblocksof

G iscalledacutvertex ofG.Noticethatifagraphconsists ofasingleblock,thenithasnocutvertex.

Theorem1([12],Theorem5.9).LetG beaminimalrealization ofaﬁnitemetricspace

(

X

,

d

)

,letG1

,

. . . ,

GkbetheblocksofG, letMibetheunionoftheverticesofX inGitogetherwiththe cutverticesofG inGi,andletdibethemetricinducedbyG on Mi.Then,ifeveryGiisanoptimalrealizationof

(

Mi

,

di

)

,thenG isalsooptimal.IfeveryGi,besidesbeingoptimal,isalsounique, thenG isoptimalanduniquetoo.

We now turn to two special classes of metrics, that is,tree metrics andcyclelikemetrics.A metricd on X is calleda treemetric if there exists an X -tree that realizes

(

X

,

d

)

,wherean X -tree isatreeT withthepropertythat eachvertexv ofT withdeg

(

v

)

≤

2 iscontainedin X [14].

Theorem2([8]).Ifd isatreemetriconaﬁnitesetX ,thenthere existsanX -treethatisauniqueoptimalrealizationof

(

X

,

d

)

.

Givena metricd on X with

|

X

|

≥

4, we saythat d is cyclelike if there is a minimal realization for d that is a cycle. Thistype of metricwas discussed ine.g., [2,12,15]. Thefollowingresultwillalsobeuseful.

Theorem3([12],Theorem4.4).Supposed isacyclelike met-riconaﬁnitesetX andacycleC isaminimalrealizationof

(

X

,

d

)

withV

(

C

)

=

X

= {

v1

,

v2

,

. . . ,

vm

}

,m

≥

4,andE

(

C

)

=

{{

vi

,

vi+1

}

:

1

≤

i

≤

m

}

,wheretheindicesaretakenmodulo

m.Then,C isanoptimalrealizationof

(

X

,

d

)

ifandonlyif

d

(

vi−1

,

vi

)

+

d

(

vi

,

vi+1

)

=

d

(

vi−1

,

vi+1

)

holdsforalli.Inthiscase,C istheuniqueoptimalrealizationof

(

X

,

d

)

.

3. Theuniquenessofoptimalrealizationsofcactus metrics

As mentioned above a cactus is a connected graph in which each edge belongs to at most one cycle. We de-ﬁne an X -cactus to bea cactus G withthe property that eachvertexv ofG withdeg

(

v

)

≤

2 iscontainedin X (see Fig. 1). Note that the maximum number of cycles in an X -cactusis

|

X

|

−

2 (whichcanbe provedbyinductionon

|

X

|

). Inaddition,wesaythat ametricd on aﬁniteset X isacactusmetric ifthereexistsanedge-weighted X -cactus thatrealizes

(

X

,

d

)

.

Givenan edge-weightedcycleC

=

v1

,

. . . ,

vm thatis a realizationofitscorrespondingmetricdC,wecallavertex vi

∈

V

(

C

)

slack ifd

(

vi−1

,

vi

)

+

d

(

vi

,

vi+1

) >

d

(

vi−1

,

vi+1

)

.

ThefollowinglemmaisadirectconsequenceofTheorem3.

Lemma4.UnderthepremiseofTheorem3,C isanoptimal re-alizationof

(

X

,

d

)

ifandonlyifC hasnoslackvertex.

Wenowusethelemmatoprovethefollowing general-izationofTheorem 2,usingtheconcept of “compactiﬁca-tion”[8,15,16].

(4)

Fig. 2. AnillustrationofcompactiﬁcationthatisdescribedintheproofofTheorem5,wherewehighlighteachslackvertexbyasquare.Compactiﬁcation ofv3intheleftgraphyieldsthegraphinthemiddlepanel,whichstillcontainsaslackvertexv4.Ifwefurtherapplythesameoperationtov4,thenwe

obtainthegraphontherightwhichhasnoslackvertex.

Theorem5.Ifd isacactusmetriconaﬁnitesetX ,thenthere existsanX -cactusthatisauniqueoptimalrealizationof

(

X

,

d

)

.

Proof. Let G bean X -cactusthatisa minimalrealization of

(

X

,

d

)

.Withoutlossofgenerality,we assumethateach cycleofG hasatleastfourvertices(sincewe canalways replace a 3-cycle witha tree insuch a waythat the ob-tained graph is a realization). If there is no cycle in G containing a slack vertex, then theassertion immediately followsfromTheorems1,3andLemma4.

So, assume that there is a cycle C

=

v1

,

. . . ,

vm in G thathasconsecutiveedges

{

vi−1

,

vi

},

{

vi

,

vi+1

}

with

i

:=

{

dG

(

vi−1

,

vi

)

+

dG

(

vi

,

vi+1

)

−

dG

(

vi−1

,

vi+1

)

}/

2

>

0.Aswe

will now explain, we apply a “compactiﬁcation” opera-tiontotheslack vertexvi (see alsoFig.2).Fornotational convenience, let

i−1

:= {

dG

(

vi−1

,

vi

)

+

dG

(

vi−1

,

vi+1

)

−

dG

(

vi

,

vi+1

)

}/

2 and

i+1

:= {

dG

(

vi+1

,

vi

)

+

dG

(

vi−1

,

vi+1

)

−

dG

(

vi

,

vi−1

)

}/

2. Compactiﬁcation of vi refers to convert-ing G into the graph G with V

(

G

)

:=

V

(

G

)

∪ {

v_i

}

and E

(

G

)

:= (

E

(

G

)

\{{

vi−1

,

vi

},

{

vi

,

vi+1

}})

∪{{

vi−1

,

vi

},

{

vi

,

vi

},

{

vi+1

,

vi

}}

, where for each j

∈ {

i

−

1

,

i

,

i

+

1

}

, the edge

{

vj

,

v_i

}

has weight

j. As can be easily veriﬁed, G is an X -cactus that is a minimal realization of

(

X

,

d

)

with a strictly smaller number ofslack vertices than G. Thus, as

|

V

(

G

)

|

isﬁnite,byapplyingthesameoperation repeat-edly andsuppressingall unlabeledverticesofdegreetwo (if anyarise), we will eventually obtain an X -cactusthat realizes

(

X

,

d

)

withouta slackvertex, whichmust be the uniqueoptimalrealizationof

(

X

,

d

)

.

2

Itisinterestingtoseethatforcactusmetrics,wedonot needtoperformtoomany“compactiﬁcations”foreach cy-cleintheaboveproofinlightofthefollowingobservation.

Proposition6.IfthepremiseofTheorem3holds,thenC hasat mosttwoslackvertices.Inthecasewhenthereexistprecisely twoslackvertices,theyareadjacentinC .

Proof. Let V

(

C

)

= {

v1

,

. . . ,

vm

}

asinTheorem 3. Suppose C hasatleasttwo slack verticesandassume that vi is a slackvertex,inotherwords,thatd

(

vi−1

,

vi

)

+

d

(

vi

,

vi+1

)

>

d

(

vi−1

,

vi+1

)

holds. As the path in C from vi−1 to vi+1

thatdoesnotcontainviistheshortestpathbetweenvi−1

and vi+1,it followsthatany v

∈

V

(

C

)

\ {

vi−1

,

vi

,

vi+1

}

is

notslack.Now,supposevi−1isaslackvertex.Thenusinga

similarargumentbyconsideringtheshortestpathbetween vi−2 and vi,itfollows that vi+1 isnotslack. Sothe only

slackverticesare vi andvi−1.Thesameargumentapplies

tothecasewhenvi+1 isaslackvertex.

2

4. Apolynomialtimealgorithmforﬁndingtheoptimal cactusrealization

In thissection we describe an algorithm, whichfor a metricd on X ,producestheuniqueoptimalrealizationfor d that isan X -cactusora messagethat there isno such realization in O

(

|

X

|

3

₎

_time. _This _should _be _compared _to

treemetricsforwhichthesameprocesscanbecarriedout inO

(

|

X

|

2

₎

_time_[₄_,₅_].

We begin by considering cyclelike metrics. Note that thecharacterization giveninTheorem 3forwhen a real-izationofacyclelikemetricisoptimalisnot suﬃcientto characterizecyclelikemetrics,aspointedoutin[15].Even sowe havethefollowingresult(whichisrelatedto Theo-rem 4.1in[2]):

Lemma7.Givenametricd onX ,wecandetermineifthereis anedge-weightedcycleC thatisanoptimalrealizationof

(

X

,

d

)

and,ifso,computeC inO

(

|

X

|

2

₎

_time.

Proof. We describe an algorithm that takes an arbitrary metricd on X as input, which incase d hasan optimal realizationthatisacyclecomputesthiscycle,andstopsif thisisnotthecase:

1)Startbyﬁndingapair

{

v0

,

v1

}

ofdistinctelementsin

X suchthatd

(

v0

,

v1

)

≤

d

(

p

,

q

)

holdsforany

{

p

,

q

}

∈

X

2

\

{{

v0

,

v1

}}

,andthensete1

:= {

v0

,

v1

}

andw1

:=

d

(

v0

,

v1

)

.

2) For each j

∈ {

2

,

. . . ,

|

X

|

−

1

}

, ﬁnd all vertices x

∈

X

\

{

v0

,

. . . ,

vj−1

}

withd

(

vj−2

,

vj−1

)

+

d

(

vj−1

,

x

)

=

d

(

vj−2

,

x

)

.

Among these vertices, we let vj be the unique vertex x thatminimizesd

(

vj−1

,

x

)

.Ifsucha vertexdoesnot exist,

or ifsuch a vertex does exist but it is not unique, then stop;elsesetej

:= {

vj−1

,

vj

}

andwj

:=

d

(

vj−1

,

vj

)

.3)Set e_|X|

:= {

v_|X|−1

,

v0

}

and w|X|

:=

d

(

v_|X|−1

,

v0

)

. 4) Check if

thecycleC deﬁnedbyV

(

C

)

:=

X andE

(

C

)

:= {

e1

,

. . . ,

e|X|

}

together withtheweight wj ofeach edge ej

∈

E

(

C

)

isa minimalrealizationof

(

X

,

d

)

.Ifnotthen stop,elseoutput theweightedcycleC .

Ifthisalgorithmreturns a cycleC that realizes

(

X

,

d

)

, thenC satisﬁestheequationinTheorem3andsoC isthe optimalrealizationof

(

X

,

d

)

.Conversely,ifthereisacycle C thatisanoptimalrealizationof

(

X

,

d

)

,thenC isunique. Inthiscase,theabovealgorithmcorrectlyconstructs C as follows. The algorithm initializes by ﬁnding two vertices

(5)

4 M. Hayamizu et al. / Information Processing Letters 157 (2020) 105916

of X that are closest together. Since an optimal realiza-tion that is a cycle is minimal, it must be the case that thesetwoverticesareconnectedbyanedge.InStep2,the algorithm iteratively extendsthe existingpath byseeking for the neighbour of vj−1, which is one of the

endver-ticesofthepath.ObservethatthetwoconditionsinStep2 uniquely determinethisneighbour:theﬁrstcondition en-sures that a shortestpathbetween vj−2 andvj contains vj−1; thesecond condition correctlyidentiﬁes the

neigh-bour of vj−1 by making sure that the distance between

it and vj−1 is shortest. In Step3, we join the two

end-vertices ofthe pathbyan edgetoformthecycleC .Note that in thisstep,we run the risk ofmaking arealization of

(

X

,

d

)

thatisapathintoarealizationof

(

X

,

d

)

thatisa cyclethat isnot minimal.Due tothis, andalsotoensure we havethecorrectsolution,wecheckthat thecycleisa minimalrealizationof

(

X

,

d

)

inStep4.

Togivetherunningtimeofthealgorithm,observethat Step1takesO

(

|

X

|

2

₎

_time_as_we_search_for_a_minimum

el-ement fromasetofsize

|X₂|

.InStep2,weiterateovera ‘forloop’atmost

|

X

|

times.Withinthe‘forloop’weiterate over at most

|

X

|

elements to find the vertices that sat-isfythefirstcondition.Then,weiterateoverthosevertices to find a minimum element from at most

|

X

|

elements. Hence, each ‘for loop’ takes O

(

|

X

|)

time; it follows then thatStep2takesO

(

|

X

|

2

)

time.Step3takesconstanttime, as we simply add a weighted edge to the graph. Since one can obtain the metricinduced by a cycle inatmost O

(

|

X

|

2

₎

_time,_Step₄_can_be_performed_in_at_most_O

₍

_|

_X

_|

2

₎

time.AseachstepofthealgorithmcanbedoneinO

(

|

X

|

2

)

time,thewholealgorithmrequiresO

(

|

X

|

2

)

time.

2

Theorem8.Givenametricd on X ,wecandetermineifd is a cactusmetricand ifsoconstructitsoptimalrealizationin O

(

|

X

|

3

₎

_time.

Proof. In [10,Algorithm2] HertzandVaronegivea poly-nomial time algorithmfordecomposingan arbitrary met-ric space

(

X

,

d

)

intoﬁnite metricspaces

(

Mi

,

di

)

,1

≤

i

≤

k, with

|

Mi

|

≤ |

X

|

, such that any optimal realization of

(

Mi

,

di

)

must consist of a single block, and such that an optimalrealizationford canbeconstructedbypiecing to-gether the optimalrealizations forthe

(

Mi

,

di

)

.Theyalso observe [10,p. 174] thatthisdecompositioncanbe com-puted in O

(

|

X

|

3

)

time using results in [7] (see also [7, p. 160]). Inaddition,bytheargumentsin[7,Lemma3.1], itfollowsthatk is O

(

|

X

|)

.

Assume that we have decomposed

(

X

,

d

)

into

{(

Mi

,

di

)

}

i∈{1,...,k}byusingtheaforementioned preprocess-ing algorithm. In case

|

Mi

|

=

2, its optimal realization is obviously a tree. Recalling the argument in the proof of Theorem 5, we know that

|

Mi

|

=

3 holds for each i

∈ {

1

,

. . . ,

k

}

.Foreach

(

Mi

,

di

)

with

|

Mi

|

≥

4,byusingthe algorithminLemma7,wecancheckif

(

Mi

,

di

)

hasan op-timalrealizationthatisacycleornot,andifsoconstruct the cyclein O

(

|

Mi

|

2

)

time (andhence O

(

|

X

|

2

)

time suf-ﬁces).Ifthereissomei

∈ {

1

,

. . . ,

k

}

suchthat

|

Mi

|

≥

4 and

(

Mi

,

di

)

doesnot haveanoptimalrealizationthatisa cy-cle,thend isnotacactusmetric,elsed isacactusmetric, and we can construct the cactus by piecing together the

optimalrealizations forthe

(

Mi

,

di

)

. Using the aforemen-tionedfact thatk is O

(

|

X

|)

,we concludethatthe overall timecomplexityisO

(

|

X

|

3

)

.

2

5. Discussionandfuturework

Itmaybeworth investigatingasto whetherthereisa moredirectandeﬃcientalgorithm thantheone givenin Theorem8forrecognizingand/or realizingcactusmetrics that usestructuralpropertiesofcactusgraphs.More gen-erally,wecouldinvestigateoptimalrealizationsformetrics that can be realized by graphs G in which every block Gi

= (

Vi

,

Ei

)

satisﬁes

|

Ei

|

− |

Vi

|

+

1

≤

k,andsuchthat ev-eryvertexin G withdegree atmost2 iscontainedin X . Here, we note that in casek

=

0, G isan X -tree, andin casek

=

1,G isan X -cactus.However,evenincasek

=

2, theremaybeinﬁnitelymanyoptimalrealizations(e.g.the metricgivenin [1,Fig.15]). So itmightbe interesting to ﬁrst understand fork

≥

2 which ofthese metrics have a unique optimalrealization, whether such metrics can be recognized inpolynomial time, and whetherthere exists apolynomial time algorithmforcomputingsome optimal realization.

Declarationofcompetinginterest

Theauthorsdeclarethat theyhavenoknown compet-ing ﬁnancialinterests orpersonalrelationships thatcould haveappeared toinﬂuence thework reportedinthis pa-per.

Acknowledgement

MH is supported by JST PRESTO Grant Number

JP-MJPR16EB,ISMCooperativeResearchProgramGrant Num-ber 2019-ISMCRP-2020, and "Challenging Exploratory Re-search Projects for the Future" grant from Research Or-ganization of Information and Systems. MH, KH and VM thanktheResearchInstituteforMathematicalSciences,an InternationalJointUsage/ResearchCenterlocatedinKyoto University,for its support.KHand VM thankthe London Mathematical Societyfor their support. KH, VM and YM thanktheNetherlandsOrganisationforScientiﬁcResearch (NWO),includingVidigrant639.072.602.

References

[1]I.Althöfer,Onoptimalrealizationsofﬁnitemetricspacesbygraphs, DiscreteComput.Geom.3 (1)(1988)103–122.

[2]A.Baldisserri,R.Elena,Distancematricesofsomepositive-weighted graphs,Aust.J.Comb.70 (2)(2018)185–201.

[3]H.-J.Bandelt,Recognitionoftreemetrics,SIAMJ.DiscreteMath.3 (1) (1990)1–6.

[4]V.Batagelj,T.Pisanski,J.M. Simões-Pereira,Analgorithmfor tree-realizability ofdistance matrices, Int. J. Comput. Math. 34 (3–4) (1990)171–176.

[5]J.C.Culberson,P.Rudnicki,A fastalgorithmfor constructingtrees fromdistancematrices,Inf.Process.Lett.30 (4)(1989)215–220.

[6]A.Dress,Trees,tightextensionsofmetricspaces,andthe cohomo-logicaldimensionofcertaingroups:anoteoncombinatorial proper-tiesofmetricspaces,Adv.Math.53 (3)(1984)321–402.

[7]A.Dress,K.T.Huber,J.Koolen,V.Moulton,A.Spillner,Analgorithm for computingcutpoints inﬁnite metric spaces,J. Classif. 27 (2) (2010)158–172.

(6)

[8]S.L.Hakimi,S.S.Yau,Distancematrixofagraphanditsrealizability, Q.Appl.Math.22 (4)(1965)305–317.

[9] M.Hayamizu,X -cactustreesandcactustreemetrics,The21stNew Zealand Phylogenomics Meeting (Waiheke 2017), 12-17 February 2017, https://cdn.auckland.ac.nz/assets/compevol/events/documents/ Waiheke2017Programme.pdf.

[10]A.Hertz,S.Varone,Themetriccutpointpartitionproblem,J.Classif. 25 (2)(2008)159–175.

[11]D.H. Huson, R.Rupp,C.Scornavacca,PhylogeneticNetworks: Con-cepts, Algorithms and Applications, Cambridge University Press, 2010.

[12]W.Imrich,J.M.Simoes-Pereira,C.M.Zamﬁrescu,Onoptimal embed-dingsofmetricsingraphs,J.Comb.Theory,Ser.B36 (1)(1984)1–15.

[13] J.Matoušek,2.7 howlargegraph? Openproblemsonembeddings of ﬁnite metric spaces workshop on discrete metric spaces and their algorithmic applications, available at http://kam.mff.cuni.cz/ ~matousek/metrop.ps,2002.

[14]C.Semple,M.Steel,Phylogenetics,OxfordUniversityPress,2003.

[15]J.M.S. Simões-Pereira, A noteon distance matrices with unicyclic graphrealizations,DiscreteMath.65 (3)(1987)277–287.

[16]J.M.S. Simões-Pereira, C.M. Zamﬁrescu, Submatrices of non-tree-realizabledistancematrices,LinearAlgebraAppl.44(1982)1–17.

[17]P.Winkler, The complexity ofmetricrealization, SIAMJ. Discrete Math.1 (4)(1988)552–559.