Distance-based vertex identification in graphs: The outer multiset dimension

ContentslistsavailableatScienceDirect

Applied

Mathematics

and

Computation

journalhomepage:www.elsevier.com/locate/amc

Distance-based

vertex

identification

in

graphs:

The

outer

multiset

dimension

Reynaldo

Gil-Pons

a

_,

_Yunior

_{Ramírez-Cruz}

b

_,

_Rolando

_{Trujillo-Rasua}

c,∗

_,

Ismael

G.

Yero

d

a Center for Pattern Recognition and Data Mining, Patricio Lumumba s/n, Santiago de Cuba 90500, Cuba

b Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, 6 av. de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg

c Centre for Cyber Security Research and Innovation, School of Information Technology, Deakin University, 221 Burwood Hwy, Burwood, VIC 3125, Australia

d Departamento de Matemáticas, Escuela Politécnica Superior de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, Algeciras 11202, Spain

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 3 March 2019 Revised 10 July 2019 Accepted 15 July 2019 Available online 5 August 2019 Keywords:

Resolvability Privacy Resolving set Multiset resolving set Metric dimension Outer multiset dimension

a

b

s

t

r

a

c

t

The characterisation of vertices in a network, in relation to other peers, has been used as a primitivein many computational procedures, such as node localisation and (de-)anonymisation.Thisarticlefocusesonacharacterisationtypeknownasthemultiset metricrepresentation.Formally,givenagraphGandasubsetofverticesS={w1,...,wt}⊆

V(G), the multiset representationof a vertex u∈V(G) with respect to S is the multiset

m(u|S)={|dG(u,w1),...,dG(u,wt)|}.AsubsetofverticesSsuchthatm(u|S)=m(v|S) ⇐⇒

u=vforeveryu,v∈V(G)\S issaidtobeamultisetresolvingset,andthecardinalityof thesmallestsuchsetistheoutermultisetdimension.Westudythegeneralbehaviourof theoutermultisetdimension,anddetermineitsexactvalueforseveralgraphfamilies.We alsoshowthatcomputingtheoutermultisetdimensionofarbitrarygraphsisNP-hard,and providemethodsforefficientlyhandlingparticularcases.

© 2019TheAuthor(s).PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBYlicense. (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Thecharacterisationofverticesina graphbymeansofunique features,knownasdistinguishabilityorresolvability,has foundapplicationsincomputernetworkswherenodesoughttobelocalisedbasedontheirpropertiesratherthanon identi-fiers[6],ortodeterminethesocialroleofanactorinsocietyincomparisontootherpeerswithsimilarstructuralproperties [7].Infact,simplestructuralpropertiesofvertices,suchastheirdegreeorthesubgraphsinducedbytheirneighbours,have beensuccessfullyusedtore-identify(supposedly)anonymoususersinsocialgraphs[8,17,18].

This article focuseson vertex characterisations that are defined in relation to a subset of vertices of the graph. The earliest of such characterisationsis known asmetric representation, introduced independently by Slater [15]in 1975and

∗ _{Corresponding author.}

E-mail addresses: rey@cerpamid.co.cu (R. Gil-Pons), yunior.ramirez@uni.lu (Y. Ramírez-Cruz), rolando.trujillo@deakin.edu.au (R. Trujillo-Rasua), ismael.gonzalez@uca.es (I.G. Yero).

https://doi.org/10.1016/j.amc.2019.124612

Hararyand Melter[3]in 1976.Formally, givenan ordered set of verticesS₌

{

w1,...,wt

}

⊆ V in a graphG=

(

V,E

)

, the metricrepresentationofavertexu∈VwithrespecttoSisthet-vectorr

(

u

|

S

)

=

(

dG

(

u,w1

)

,...,dG

(

u,wt

))

,wherethemetric

dG

(

u,

v

)

is computedasthe length ofa shortest u−

v

path in G. An ordered subset S satisfyingthat every two distinct vertices u and

v

in the graph have different metric representation, i.e.r

(

u

|

S

)

_=r

(

v

|

S

)

_, is said to be a resolving set. The minimumcardinalityamongsttheresolvingsetsinagraphGisknownasthemetricdimensionofG,anddenotedasdim

(

G

)

. The metric dimension of graphs has beenextensively studied in literature since the 70s. Issues that are relevant to the presentday,suchasprivacyinonlinesocialnetworks,arestillbenefitingfromsuchresearcheffort[9–12,16].

Theassumptionthatresolvabilityrequiresan ordertoexist(orbeimposed)onasetSforobtainingmetric representa-tionsremainedunchallengeduntil2017,whenSimanjuntak,Vetrík,andMuliaintroducedthenotionofmultiset representa-tion[14]bylookingatthemultisetofdistancesratherthanatthestandardvectorofdistances.

Foravertexu∈VandavertexsetS⊆ V,themultisetrepresentationofuwithrespecttoS,denotedm(u|S),isdefinedby

m

(

u

|

S

)

=

{|

dG

(

u,w1

)

,...,dG

(

u,wt

)

|}

, where

{|

.

|}

denotesamultiset.

Usingthisdefinition,thenotionsofresolvabilityintermsofthemetricrepresentationwerestraightforwardlyextended to consider resolvabilityinterms ofthe multiset representation[5,14].Ourmain observation inthis articleis that these straightforward extensionsarein factan oversimplificationofthe problemofdistinguishing verticesina graphbasedon themultisetrepresentation.Wearguethatthisproblemhastwoflavours,oneofwhichhasbeenneglectedinliterature.

Contributions.Thisarticlemakesthefollowingcontributions.

• We generalisethemetricdimension ofgraphs toaccommodate differentcharacterisationsoftheir vertices,such as themetric andmultisetrepresentations.We showthat themetric dimensionproblemwithrespectto themultiset representationadmitstwointerpretations:onethatcanbefoundintheliterature[5,14]andisknownasthemultiset dimension,andanotheronethatwecalltheoutermultisetdimension.Thelatteriswell-defined,whereasthemultiset dimensionisundefinedforaninfinitenumberofgraphs[5,14].Wealsoshowthattheoutermultisetdimensionfinds applicationsonmeasuringthere-identificationriskofusersinasocialgraph.

• Wecharacteriseseveralgraphfamiliesforwhichtheoutermultisetdimensioncanbeeasilydetermined,orbounded bythemetricdimension.

• WeprovethattheproblemofcomputingtheoutermultisetdimensioninagraphisNP-Hard.

• Weprovideapolynomialcomputationalproceduretocalculatetheoutermultisetdimensionoffull2-arytrees,and aparallelisablealgorithmforthegeneralcaseoffull

δ

-arytrees.

Structureofthearticle.InSection2,wediscussthegeneralisationofthenotionofmetricdimension,focusingonvector andmultisetmetricrepresentationsasparticularcasesofinterest.FromSection3onwards,thepaperfocusesontheouter multisetdimension. Section 3isdevoted tothe basicpropertiesofthisparameter, whereas Section4 discussesthe com-plexityofitscomputation.Finally,Section5studiesthebehaviouroftheoutermultisetdimensionintheparticularcaseof trees.

2. Ageneralisationofthemetricdimension

We considera simple andconnectedgraphG=

(

V,E

)

whereV isa set ofverticesandE a set ofedges. The distance

dG

(

v

,u

)

between two vertices

v

and u in G is the number of edges ina shortest path connecting them. Ifthere is no ambiguity,wewillsimplywrited

(

v

,u

)

.

Themetricdimensionofgraphshastraditionallybeenstudiedbasedontheso-calledmetricrepresentation,whichisthe vector ofdistances fromavertexto anordered subset ofverticesof thegraph.Toaccommodate other typesofrelations between vertices,we generalise the metricdimension by considering any equivalence relation ∼ ⊆ V× Vover the set of verticesofthegraph.Thatis,weconsiderarelation∼ thatisreflexive,symmetric,andtransitive.Weuse[u]_∼todenotethe equivalenceclassofthevertexu∈Vwithrespecttotherelation ∼ ,whileV/∼ denotes thepartition ofV composedofthe equivalenceclassesinducedby∼.

Definition2.1(Resolvingandouter resolvingset). AsubsetSofverticesinagraphG=

(

V,E

)

issaidtoberesolving(resp. outerresolving)withrespectto∼ ifallequivalenceclassesinV/∼ (resp.

(

V− S)/∼)havecardinalityone.

Whilestandardresolvingsetsdistinguishallverticesinagraph,outerresolvingsetsonlylookatthoseverticesthatare notinS,hencethename.Weremarkthatthereexistapplicationsworkingundertheassumptionthat Sisgiven,implying that vertices in S donot needto be distinguishable.For example,in an active re-identificationattack on a social graph [1,11,16],amaliciousagent,theattacker,firstinsertsasetoffakeaccountsinthegraph,commonlycalledsybils,andcreates asetofuniqueconnectionpatternswithanumberoflegitimatevertices,calledvictimsortargets.Afterasanitisedversion ofthesocialgraphisreleased,theattackerretrievesthesetofsybilnodesbyusingapatternmatchingalgorithm,andthen re-identifiesthevictimsbymeansontheirmetricrepresentationswithrespecttothesetofsybils.

multiset resolvability

outer multiset resolvability

outer resolvability

resolvability

Fig. 1. Hierarchy of resolvability notions.

Proposition2.2. Foreverynon-trivialgraphG,thefollowingfactshold:

i. EveryresolvingsetofGwithrespectto∼₌Sisanouterresolvingset.

ii. EveryouterresolvingsetofGwithrespectto∼₌S isanouterresolvingsetofGwithrespectto ∼S. iii. EveryouterresolvingsetofGwithrespectto ∼S isaresolvingsetofG,andviceversa.

Proof.Let S⊆ V(G) be a resolving set of G with respect to ₌∼S. Then, every pair of distinct vertices u,

v

∈V

(

G

)

satisfy m

(

u

|

S

)

=m

(

v

|

S

)

.Thus, it triviallyfollows that the sameproperty holds forevery pair of distinct verticesu,

v

∈V

(

G

)

\

S.

Thiscompletestheproofof(i).

Thesecondpropertyfollowsstraightforwardlyfromthefactthatm

(

u

|

S

)

=m

(

v

|

S

)

⇒r

(

u

|

S

)

=r

(

v

|

S

)

,and(iii)isa

well-knownpropertyofresolvingsetsbasedonthemetricrepresentation. 

Fig.1depictstherelationsbetweenresolvabilitynotionsenunciatedinProposition2.2intheformofahierarchy.Inthe figure,every arrowfromresolvabilitynotionAtoresolvabilitynotionBindicatesthatasetSwhichisresolvingasdefined byAisalsoresolvingasdefinedbyB.WeusethefollowingshorthandnotationinFig.1andintheremainderofthisarticle.

• resolvingsettodenotearesolvingsetwithrespectto ∼S. • multisetresolvingsettodenotearesolvingsetwithrespectto∼₌S. • outerresolvingsettodenoteanouterresolvingsetwithrespectto ∼S. • outermultisetresolvingsettodenoteanouterresolvingsetwithrespectto∼₌S.

Definition2.3(Metric dimension andouter metricdimension). Themetricdimension (resp.outer metricdimension)ofa simpleconnectedgraph G₌

(

V_,E

)

withrespecttoa structuralrelation_{∼ is} theminimumcardinality amongst aresolving (resp.outer resolving)setinGwithrespectto ∼ .Ifnoresolving(resp.outer resolving)setexists,we saythatthemetric dimension(resp.outermetricdimension)isundefined.

AnexampleofametricdimensiondefinitionthatisundefinedforsomegraphsisgivenbySimanjuntaketal.[14].They usethemultisetrepresentationtodistinguishvertices.Itiseasytoprovethat acompletegraphhasnomultisetresolving set,whichleadstoindefinition.Conversely,theoutermetricdimensionwithrespecttothemultisetrepresentationisalways defined,giventhatforeverygraphG=

(

E,V

)

,Visanoutermultisetresolvingset.

Overall,wehighlightthefactthat,whiletheoutermetricdimensionandthestandardmetricdimensionwithrespectto themetricrepresentationareequivalent(seeFig.1),theuseofthemultisetrepresentationrenderstheoutermetric dimen-siondifferentfromthestandardmetricdimension.Infact,theoutermultisetdimensionisdefinedforanygraph,whereas themultisetdimensionisnot.Furthermore,recentprivacyattacksandcountermeasuresonsocialnetworks[1,11,13,16]rely onthenotionofouterresolvingset,ratherthanontheoriginalnotionofresolvingset.Theremainderofthisarticleisthus dedicatedtothestudyoftheoutermultisetdimension,thatis,theoutermetricdimensionwithrespectto∼₌S.

3. Basicresultsontheoutermultisetdimension

Inthissectionwecharacterise severalgraphfamiliesforwhichtheoutermultisetdimensioncanbeeasily determined, orbounded by the metric dimension otherwise. We start by providing notation that we use throughout therest of the paper.

Notation.LetG=

(

V,E

)

bea simpleundirectedgraphofordern=

|

V

(

G

)

|

.We willsaythatG isnon-trivialifn≥ 2.We denotebyKn,Nn,PnandCn thecomplete,empty,pathandcyclegraphs,respectively,ofordern.Moreover,wewillusethe notationu_↔G

v

(negatedasu↔G

v

)toindicatethatuand

v

areadjacentinG,thatis

(

u,

v

)

∈E.Foravertex

v

ofG,NG

(

v

)

denotesthesetofneighboursof

v

inG,thatisNG

(

v

)

=

{

u∈V

(

G

)

: u↔

v

}

.ThesetNG

(

v

)

iscalledtheopenneighbourhood ofthevertex

v

inGandNG[

v

]=NG

(

v

)

∪

{

v

}

iscalledtheclosedneighbourhoodof

v

inG.Thedegreeofavertex

v

ofGwill bedenotedby

δ

G

(

v

)

.Ifthereisnoambiguity, wewill dropthesubscriptsandsimplywrite u↔

v

, u↔

v

,N

(

v

)

,etc. Two differentverticesu,

v

arecalledtruetwinsifN[u]=N[

v

].Likewise, u,

v

arecalledfalsetwinsifN

(

u

)

=N

(

v

)

.Ingeneral,u,

v

Finally,wewillusethenotationdimms(G)fortheoutermultisetdimensionofagraphG,anddim

(

G

)

forthestandardmetric

dimension.

Proposition3.1. Foreverynon-trivialgraphGofordern,thefollowingfactshold: i.1≤ dimms

(

G

)

≤ n− 1.

ii.dimms

(

G

)

≥ dim

(

G

)

.

Proof. Thefactthatdimms(G)≥ 1followsdirectlyfromthedefinitionofoutermultisetdimension,whereasdimms

(

G

)

≤ n− 1

followstriviallyfromthefactthateveryvertex

v

isthesolevertexinV

(

G

)

\

(

V

(

G

)

\

{

v

}

)

,andthusithasauniquemultiset representation w.r.t.V

(

G

)

\

{

v

}

, which is thus a multiset resolving set. The fact that dimms

(

G

)

≥ dim

(

G

)

follows directly

fromitems(ii)and(iii)ofProposition2.2. 

Onceestablished theglobalboundsof theouter multisetdimension, wenow focuson theextremecasesofthese in-equalities.

Proposition3.2. AgraphGsatisfiesdimms

(

G

)

=1ifandonlyifitisapathgraph.

Proof. LetGbeapathgraph.Itisclearthattheset

{

v

}

,where

v

isanextremevertexofG,isamultisetresolvingsetofG, sodimms(G)≤ 1.Byitem(ii)ofProposition3.1,dimms

(

G

)

≥ dim

(

G

)

≥ 1,sotheequalityholds.Ontheotherhand,ifGisnot

apathgraph,thenitem(ii)ofProposition3.1alsoleadstodimms

(

G

)

≥ dim

(

G

)

≥ 2,asthestandardmetricdimensionofa

graphisknowntobe1ifandonlyifitisapathgraph[2]. 

According to Proposition 3.2, the caseswhere dimms

(

G

)

=dim

(

G

)

=1 coincide. However, thisis not thecase forthe

upperbound ofProposition3.1(i). Indeed,whileit iseasy tosee that,foranypositive integern_{≥ 2,}thecomplete graph

Knsatisfiesdimms

(

Kn

)

=dim

(

Kn

)

=n− 1,wehavethefactthatthisisthesolefamilyofgraphsforwhichdim

(

Kn

)

=n− 1, whereasthereexistgraphsGsuchthatdimms

(

G

)

=n− 1>dim

(

G

)

,asexemplifiedbythenextresults.

Example3.3. ThecyclegraphsC4andC5 satisfydimms

(

C4

)

=3>2=dim

(

C4

)

anddimms

(

C5

)

=4>2=dim

(

C5

)

.

Proposition 3.4. Every complete k-partite graph G∼₌Kr₁,r2,...,rk such that r1=r2=· · · =rk≥ 2 and k

i=1ri=n satisfies dimms

(

G

)

=n− 1.

Proof. LetG∼₌Kr₁,r2,...,rk beacompletek-partitegraphsuchthatr1=r2=· · · =rk≥ 2.Letu,

v

∈V

(

G

)

betwoarbitrary ver-ticesofGandletS⊆ V

(

G

)

\

{

u,

v

}

.Ifu↔

v

,thenm

(

u

|

S

)

=m

(

v

|

S

)

,astheyarefalsetwinsinG.Consequently,Sisnota multisetresolvingsetofG.Wenowtreatthecasewhereu_↔

v

,forwhichwedifferentiatethefollowingsubcases:

• S=V

(

G

)

\

{

u,

v

}

.In thiscase, m

(

u

|

S

)

=m

(

v

|

S

)

=r−1 i=1

{|

2

|}

∪ r−1 i=1

{|

1

|}

∪ k−2 i=1 r

j=1

{|

1

|}

, andsoS is nota multiset

resolvingsetofG.

• S⊂ V

(

G

)

\

{

u,

v

}

. Here, if there exists some x∈V

(

G

)

\

(

S∪

{

u,

v

}

)

such that x↔u (resp. x↔

v

), then m

(

u

|

S

)

= m

(

x

|

S

)

(

resp.m

(

v

|

S

)

₌m

(

x

|

S

)

)

_, asx and u(resp. x and

v

) are false twins inG. Thus, S is not a multiset re-solvingsetofG.Finally,ifeveryx∈V

(

G

)

\

(

S∪

{

u,

v

}

)

satisfiesu↔x↔

v

,thenwe havethat m

(

u

|

S

)

=m

(

v

|

S

)

= r−1

i=1

{|

2

|}

∪ri=1−1

{|

1

|}

∪ti=11

{|

1

|}

∪· · · ∪

t_k−2

i=1

{|

1

|}

, withti≤ r for i∈

{

1,...,k− 2

}

, which entailsthat S isnot a multiset resolvingsetofG.

Summingupthecasesabove,wehavethatnosetS⊆ V(G)suchthat

|

S

|

≤ n− 2isamultisetresolvingsetofG,andso dimms

(

G

)

≥ n− 1.Theequalityfollowsfromitem(i)ofProposition3.1.Theproofisthuscompleted. 

Example3.3showstwo caseswhere theouter multiset dimensionof acyclegraph isstrictly larger thanits standard metricdimension.Withtheexception ofC3,whichsatisfiesdimms

(

C3

)

=dim

(

C3

)

=2,thestrict inequalityholdsforevery

othercyclegraph,asshownbythefollowingresult.

Proposition3.5. EverycyclegraphCn ofordern≥ 6satisfiesdimms

(

Cn

)

=3.

Proof. Consider anarbitrarypair ofverticesu,

v

∈V

(

Cn

)

anda pairofverticesx,y∈V

(

Cn

)

\

{

u,

v

}

such that u↔x,y↔

v

, and both x and y lie on exactly one path from u to

v

(note that for n≥ 6 atleast one such pair x, y exists). We have that m

(

x

|

{

u_,

v

}

)

=m

(

y

|

{

u_,

v

}

)

=

{|

1_,d

(

u_,

v

)

± 1

|}

, sono vertex subset of size 2is a multiset resolving set ofCn. Thus, dimms(Cn)≥ 3.

Now,consideranarbitraryvertex

v

i∈V

(

Cn

)

andthesetS=

{

v

i−2,

v

i,

v

i+1

}

,wherethesubscriptsaretakenmodulon.We

differentiatethefollowingcasesforapairofverticesx,y∈V(Cn)ࢨS:

1. x₌

v

i−1.Inthiscase,m

(

x

|

S

)

=

{|

1,1,2

|}

=m

(

y

|

S

)

,asyisatdistance1fromatmostoneelementinS.

2. x andy satisfy

{

d

(

x,

v

i

)

,d

(

x,

v

i−2

)

}

=

{

d

(

y,

v

i

)

,d

(

y,

v

i−2

)

}

.In thiscase, assuming without lossof generality that a= d

(

x,

v

i

)

<d

(

y,

v

i

)

,wehavethatm

(

x

|

S

)

=

{|

a,a+2,a− 1

|}

=

{|

a,a+2,a+3

|}

=m

(

y

|

S

)

.

3. x and y satisfy

{

d

(

x_,

v

i+1

)

,d

(

x,

v

i−2

)

}

=

{

d

(

y,

v

i+1

)

,d

(

y,

v

i−2

)

}

. In a manner analogous to that of the previous case,

we assume without loss of generality that b=d

(

x,

v

i+1

)

<d

(

y,

v

i+1

)

and obtain that m

(

x

|

S

)

=

{|

b,b+1,b+3

|}

=

{|

b,b+2,b+3

|}

=m

(

y

|

S

)

.

Fig. 2. The wheel graph W 1 ,5 ∼= v + C 5 .

Finally,summing upthe casesabove, we havethat S isa multisetresolving set ofG,andso dimms

(

G

)

≤

|

S

|

=3.This

completestheproof. _

Next,we characterise a large number of cases where the outer multiset dimension is strictly greater than the stan-dard metric dimension. To that end, we first introduce some necessary notation. We represent by n

rCrep the number of

r-combinations,withrepetition,fromnelements. Likewise,werepresentbyn

rPrepthenumberofr-permutations,with rep-etition, from n elements. Recall that n

rCrep=



r+nr−1



=



r+n−1 n−1



, whereas n

rPrep=nr. Finally, we recall the quantity f(n, d), definedin [2]as the smallest positive integer k such that k₊dk_{≥ n. In an analogous manner, we define f}_{(n, d) asthe} smallestpositiveintegerk suchthatk+



r+d−1

d−1



≥ n.Since,bydefinition,n

rCrep≤nrPrep,wehavethat f(n,d)≤ f(n,d). With thepreviousdefinitionsinmind,weintroduceournextresult.

Theorem3.6. ForeverygraphG=

(

V,E

)

ofordernanddiameterdsuchthatdim

(

G

)

< f

(

n,d

)

, dimms

(

G

)

>dim

(

G

)

.

Proof.LetG=

(

V,E

)

bea graphofordernanddiameterd.Itwasproven in[2]thateverysuch graphsatisfiesdim

(

G

)

≥

f

(

n,d

)

.Indeed,novertexsubsetS⊆ Vsuchthat|S|<f(n,d)isametricgeneratorofG,becausethenumberofdifferentmetric representations,withrespecttoS,forelementsinV_ࢨS isatmostd|S|_<n−

_|

_S

_|

₌

_|

_V

_\

_S

_|

_{.Ingeneral,if}

_|

_S

_|

_{=r,thesetofall} possibledifferentmetricrepresentations forelementsofVࢨSwithrespecttoSisthatofallpermutations, withrepetition, ofr elementsfrom

{

1,2,...,d

}

.Applying an analogousreasoning,we have that the setof all possible differentmultiset metricrepresentationsforelementsofV_ࢨSwithrespecttoSisthatofallcombinations,withrepetition,ofrelementsfrom

{

1,2,...,d

}

.Thus,anymultisetmetricgeneratorSofGmustsatisfy n

|S|Crep≥ n−

|

S

|

,sodimms(G)≥ f(n,d).Inconsequence,

ifdim

(

G

)

< f

(

n,d

)

,thendimms

(

G

)

>dim

(

G

)

. 

AnexampleofthepreviousresultisthewheelgraphW1,5∼=



v



+C5,whichhasdiameter2(seeFig.2).Asdiscussedin

[2,4],dim

(

W1,5

)

=2= f

(

6,2

)

< f

(

6,2

)

=3<4=dimms

(

W1,5

)

.

Toconcludethissection,wegiveageneralresultontherelationbetweenoutermultisetresolvingsetsandtwinvertices, aparticularcaseofwhichwillbeusefulinfurthersectionsofthispaper.

Proposition3.7. LetGbea non-trivialgraphandletS_{⊆ V}(G)beanoutermultisetresolvingsetofG.Letu_,

v

∈V

(

G

)

beapair oftwinvertices.Then,u∈Sor

v

∈S.

Proof.The proof follows fromthe fact that, astwin vertices,u and

v

satisfy d

(

u,x

)

=d

(

v

,x

)

for every x∈V

(

G

)

\

{

u,

v

}

,

whichentailsthatuand

v

havethesamemultisetrepresentationaccordingtoanysubsetofV

(

G

)

\

{

u,

v

}

. 

Corollary3.8. LetGbeanon-trivialgraphandlet_{T =}

{

[u1],[u2],...,[ut]

}

bethesetofequivalenceclassesinducedinV(G)by

thetwinequivalencerelation.Then, dimms

(

G

)

≥

t



i=1

(

|

[ui]

|

− 1

)

.

Proof.The resultfollowsfromthe factthat,forevery twinequivalence class,atmostone elementcan beleft outofany

outermultisetresolvingset. 

4. Complexityoftheoutermultisetdimensionproblem

Intheprevious section,we showedthatalgorithms abletocompute themetric dimensioncanbe usedto determined orbound theoutermultisetdimension.Thetroubleis,however,thatcalculatingthemetricdimensionisNP-Hard[6].We prove inthis section that computingthe outer multiset dimension ofa simple connected graphis NP-hard as well. The proofis,insome way,inspiredbytheNP-hardnessproofofthemetricdimensionproblemgivenin[6].Tobeginwith, we formallystatethedecisionproblemassociatedtothecomputationoftheoutermultisetdimension:

OuterMultisetDimension(DimMS)

Fig. 3. Gadget of a variable x i .

Fig. 4. Gadget of clause C j .

Theorem4.1. TheproblemDIMMSisNP-complete.

Proof. TheproblemisclearlyinNP. Wegive theNP-completenessproof byareduction from3-SAT. Consideran arbitrary inputto3-SAT,thatis,aformulaFwithnvariablesandmclauses.Letx1,x2,...,xnbethevariables,andletC1,C2,...,Cmbe theclausesofF.WenextconstructaconnectedgraphGbasedonthisformulaF.Tothisend,weusethefollowinggadgets.

Foreachvariablexiweconstructagadgetasfollows(seeFig.3).

• NodesTi,Fiarethe“true” and“false” endsofthegadget.Thegadgetisattachedtotherestofthegraphonlythrough thesenodes.

• Nodesa1

i,a2i,b1i,b2i “represent” thevalue of the variablexi,that is,a1i and a2i will be used torepresentthatvariable

xiistrue,andb1i andb2i thatitisfalse. • Nodesd1

i andd2i willhelptodifferentiatebetweennodesindifferentgadgets. • Q_i is aset ofend-nodes of cardinalityq_iadjacent to d1

i. Notice that all thesenodesare indistinguishablefrom di2. Moreover,thecardinalitiesofthesesetsQiarepairwisedistinct,whichisnecessaryforourpurposesintheproof.We furtheronstatetheexplicitvaluesoftheircardinalities.

ForeachclauseCjweconstructagadgetasfollows(seeFig.4). • Nodesc1

j andc3j willbehelpfulindeterminingthetruthvalueofCj. • Nodesc2

j andc4j willhelptodifferentiatebetweennodesindifferentgadgets.

• Pj isasetofend-nodesofcardinalitypjadjacenttoc2j.Noticethatallthesenodesareindistinguishablefromc4j.As inthecaseofthesetsQ_ifromthevariablegadgets,thecardinalitiesofthesesetsP_jarealsopairwisedistinct. Asmentionedbefore,werequiresomeconditionsonthecardinalitiesofthesetsPiandQifromthevariablesandclauses gadgets,respectively.Thevaluesoftheircardinalities(whichwerequireinourproof)areasfollows.Foreveryi∈

{

1,...,n

}

we make qi=2· i· n,and forevery j∈

{

1,...,m

}

we make pj=2· j· n+2n2. In concordance, we notice that theset of numberspiandqjarepairwisedistinct.Also,weclearlyseethatqi+pjispolynomialinn+m.

Thegadgetsrepresentingthevariables andthe gadgetsrepresentingthe clausesare connectedinthe followingwayin ordertoconstructourgraphG.

• Nodesc1

j,foreveryj,areadjacenttonodesTi,Fiforalli.

• IfavariablexidoesnotappearinaclauseCj,thenthenodesTi,Fiareadjacenttoc3j. • Ifavariablex_iappearsasapositiveliteralinaclauseC_j,thenthenodeF_iisadjacenttoc3

Wefirstremarkthat theconstructedgraphGisconnected,andthatits orderispolynomial inthenumberofvariables andclausesofthe original3-SAT instance.Wewillprove nowthat theformulaFissatisfiable ifandonlyifthemultiset dimensionofGisexactlyM=n

i=1qi+mj=1pj+n.

First,let uslook atsome propertiesthat must be fulfilledby a multisetresolving set Sof minimumcardinality inG. First,asthenodesinQi∪

{

d2i

}

,foreveryi∈

{

1,...,n

}

,areindistinguishableamongthem,andatleast|Qi|ofthemmustbe inS,wecanassume withoutlostofgeneralitythat Qi⊂ S.Byusingasimilarreasoning,alsoPj⊂ Sforevery j∈

{

1,...,m

}

. Moreover,foreveryi∈

{

1,...,n

}

,atleastoneofthenodesa1

i,a2i,b1i,b2i mustbeinS,otherwisesomepairsofthemwould havethesamemultisetrepresentation,whichisnotpossible.Thus,thecardinalityofSisatleastM.Clearly,ifM=

|

S

|

,then wehavealreadyfullydescribedasetofnodesthatcouldrepresentS.

Lemma4.2. Consider a set S∗ containingexactly M nodesgiven as follows. Allnodes in Qi fori∈

{

1,...,n

}

,all nodesin Pj

for j∈

{

1,...,m

}

,and exactlyonenodefrom eachset

{

a1

i,a2i,b1i,b2i

}

fori∈

{

1,...,n

}

arein S∗.Then, allpairsofnodeshave

differentmultisetrepresentationswithrespecttoS∗,exceptpossiblyc1

j andc3j (forsome j∈

{

1,...,m

}

).

Proof.Toprovethelemma,wewillexplicitlycomputethemultisetrepresentationofeachnode.Foreasierrepresentation, weuseavector

(

x1,. . .,xn

)

todenotethemultisetoverpositiveintegerssuchthat 1hasmultiplicityx1,2hasmultiplicity x2,andsoon. • m

(

c4 j

|

S∗

)

=

(

0,pj,0,n,...

)

, • m

(

c2 j

|

S∗

)

=

(

pj,...

)

, • m

(

d1 i

|

S∗

)

=

(

qi,...

)

, • m

(

d2 i

|

S∗

)

=

(

0,qi,...

)

, • (m(Ti|S∗),m(Fi|S∗))isequaltoeither

((

1,qi,...

)

,

(

0,qi+1,...

))

or

((

0,qi+1,...

)

,

(

1,qi,...

))

, • a1

i,a2i,b1i,bi2:Let’sassumeb2i ∈S∗.Then

{

m

(

a1i

|

S∗

)

,m

(

a2i

|

S∗

)

,m

(

bi1

|

S∗

)

}

=

{

(

1,0,qi,...

)

,

(

0,1,qi,...

)

,

(

0,0,qi+1,...

)

}

. Ananalogousresultremainsiftheassumptionthatb2

i ∈S∗isdropped,basedonthefollowingobservations.First,one andonlyoneofthenodesa1

i,a 2 i,b 1 i,b 2

i isinS∗,andthedistancesfromtheotherthreetothisoneareexactly1,2,3 insomeorder.Second,eachofthesenodeshaveqinodesatdistance3.

• m

(

c1

j

|

S∗

)

=

(

0,pj+n,ni=1qi,ml=1pl− pj

)

, • c3

j:thenumberofnodesatdistancetwodependsonwhichnodebelongstoS∗fromeachvariablegadget.We distin-guishthreepossiblecasesforthedistancebetweenc3

j andthenodefromS∗belongingtothegadgetcorresponding toavariablexi.

– IfxiappearsinCjasapositiveliteral,anda1i ∈S∗ora2i ∈S∗,thensuchdistanceis3. – IfxiappearsinCjasanegativeliteral,andb1i ∈S∗orb2i ∈S∗,thensuchdistanceis3. – Ifnoneoftheabovesituationsoccurs,thensuchdistanceis2.

Therefore,themultisetrepresentation ofc3

j isrelatedto theset

(

0,pj+wj,qi+n− wj,pl− pj

)

, wherewj isthe numberofnodesfromgadgetsrepresentingsome ximatchingthethirdcaseabove.Noticethat,asthedifferencebetween anypjandanyqiisatleast2n,allpairsofnodeshavealsoadifferentmultisetrepresentation,exceptpossibly

(

c1j,c3j

)

that dependontheselectednodesfromeachvariablegadget.Wenextparticularisesomeofthesesituations.

• Asqi=pjforeveryi∈

{

1,...,n

}

andevery j∈

{

1,...,m

}

,weobservem

(

c2j

|

S∗

)

=m

(

di1

|

S∗

)

,m

(

c4j

|

S∗

)

=m

(

d2i

|

S∗

)

. • Since pj=qi+1 for every i∈

{

1,...,n

}

and every j∈

{

1,...,m

}

, we deduce m

(

c4j

|

S∗

)

=m

(

Ti

|

S∗

)

and m

(

c4j

|

S∗

)

=

m

(

Fi

|

S∗

)

.

• Sincepj1=pj2+nforevery j1,j2∈

{

1,...,m

}

,wegetm

(

c 4

j1

|

S

∗

₎

_=m

₍

_c1

j2

|

S ∗

₎

_.

Remainingcasestriviallyfollow,andarelefttothereader,andsotheproofofthelemmaiscomplete.  Wewill nowshow awayto transformthesetS∗ intovaluesforthe variablesxi thatwill leadto asatisfiable assign-ment for F. If S∗_∩

{

a1

i,a

2

i

}

=∅ for some variable xi, then we set the variable xi=true (with respect to S∗). Otherwise (S∗∩

{

a1

i,a2i

}

=∅or equivalentlyS∗∩

{

b1i,b2i

}

=∅), we set xi=false.Hence, theclause Cj istrue or falseinthe natural way,accordingtothevaluespreviouslygiventoitsvariables.

Lemma4.3. LetS∗beasetofnodesasdefinedinthepremiseofLemma4.2.Thenc1

j andc3j havedifferentmultiset

representa-tionswithrespecttoS∗ifandonlyiftheclauseCjistrue.

Proof.Notice thatthedistancebetweenc3

j andthenode inS∗ fromthegadgetcorresponding toxiis3ifandonlyifthe clauseCjistrue(seeLemma4.2).Thus,wj=0(asdefinedinLemma4.2)whentheclauseCjisfalse,andonlyinthiscase m

(

c3

j

|

S∗

)

=m

(

c1j

|

S∗

)

. 

Byusingthelemmasabove,weconcludetheNP-completenessreduction,throughthefollowingtwolemmas.

Proof. Recallthatdimms(G)≥ M.ItremainstoprovethatifFissatisfiablethendimms(G)≤ M.LetusconstructasetSinthe

followingway. Ifxiistrue,then a1i ∈S.Otherwise (xi isfalse),b1i ∈S.Also,we add toS allnodesinthe setsPj andQi. Hence,accordingtoLemmas4.2and4.3,Sisamultisetresolvingset,anditscardinalityisexactlyM. 

Lemma4.5. IftheoutermultisetdimensionofGisM,thenFissatisfiable.

Proof. LetS be a set ofnodes of cardinality equal to the multisetdimension ofG. Hence, as explained before,without lostofgeneralityall nodesinthesetsPj,Qi,andexactlyonenode ofa1i,a2i,b1i,bi2,mustbelongtoS,andnoothernode isin S. Ifa1

i ∈S ora2i ∈S, thenlet xibe true.Otherwise, letxi befalse.Since Sisa multisetresolving set,accordingto Lemmas4.2and4.3,allclausesCjofFmustbetrue,unlessthenodesc1j andc3j wouldhavethesamemultiset representa-tion,whichisnotpossible.IfallclausesofFaretrue,thenFissatisfiable,asclaimed.  Thelasttwolemmastogethercompletethereductionfrom3-SATtotheproblemofdecidingwhethertheoutermultiset dimensionofagraphGisequaltoa givenpositive integer.Thelatterproblemcaninturnbe triviallyreducedtoDimMS.

Thiscompletestheproof. 

5. Particularcasesinvolvingtrees

Giventhat,ingeneral, computingtheouter multisetdimensionofa graphisNP-hard,itremains anopen questionfor whichfamiliesofgraphstheouter multisetdimensioncanbeefficientlycomputed.Thegoalofthissectionistoprovidea computationalprocedureandaclosedformulatocomputetheoutermultisetdimensionoffull

δ

-arytrees.Afull

δ

-arytree

isarootedtreewhoseroothasdegree

δ

,allitsleavesareatthesamedistancefromtheroot,anditsdescendantsareeither leavesorverticesofdegree

δ

+1.Weexpecttheresultsobtainedinthissectiontopavethewayforthestudyoftheouter multisetdimensionofgeneraltrees.

Notation.Givena multisetM andan elementx, wedenote themultiplicity ofxinM asM[x].We use



G(x) to denote theeccentricityofthevertexxinagraphG,whichisdefinedasthelargestdistancebetweenxandanyothervertexinthe graph.Wewillsimply write



(x)iftheconsidered graphisclearfromthe context.Givena treeTrootedinw,weuseTx todenotethesubtreeinducedbyxandalldescendantsofx,i.e.thoseverticeshavingashortestpathtowthat containsx. Finally,anoutermultisetbasisisanoutermultisetresolvingsetofminimumcardinality.

Westartbyenunciatingasimplelemmathatcharacterisesmultisetresolvingsetsinfull

δ

-arytrees.

Lemma5.1. LetTbeafull

δ

-arytreerootedinwwith

δ

_>1.A setofverticesS_{⊆ V}(T) isanoutermultisetresolvingsetifand onlyif

∀

u,v∈V(T)\S:d

(

u,w

)

=d

(

v

,w

)

⇒m

(

u

|

S

)

=m

(

v

|

S

)

.

Proof. Necessityfollowsfromthedefinitionofoutermultisetresolvingsets.Toprovesufficiencyweneedtoprovethat

∀

u,v∈V(T)\S:d

(

u,w

)

=d

(

v

,w

)

⇒m

(

u

|

S

)

=m

(

v

|

S

)

.

Take two vertices x, y∈V(T)ࢨS such that d

(

x,w

)

<d

(

y,w

)

. Because T is a full

δ

-ary tree, we obtain that d

(

x,w

)

< d

(

y_,w

)

_⇐⇒



T

(

x

)

<



T

(

y

)

. Also, there must exist two leaf vertices y1, y2 inT which are siblings and satisfy d

(

y1,y

)

= d

(

y2,y

)

=



T

(

y

)

.Consideringthaty1andy2 arefalsetwins,weobtainthaty1,y2∈S⇒m

(

y1

|

S

)

=m

(

y2

|

S

)

.Therefore,given

thatd

(

y1,w

)

=d

(

y2,w

)

,itfollowsthaty1∈Sory2∈S.Weassume,withoutlossofgenerality,thaty1∈S.Ontheonehand,

wehavethatd(y1,y)∈m(y|S).Ontheotherhand,becaused

(

y1,y

)

=



T

(

y

)

>



T

(

x

)

,weobtainthatd(y1,y)∈m(x|S),implying

thatm(x|S)=m(y|S). 

Basedontheresultabove,weprovideconditionsunderwhichanoutermultisetbasiscanbeconstructedinarecursive manner. This is usefulforthe developmentof a computational procedure that finds the outer multiset dimension of an arbitraryfull

δ

-arytree.

Lemma5.2. Givena naturalnumber>1,let T1,...,Tδ be

δ

full

δ

-arytrees ofdepthwithpairwisedisjoint vertexsets. Let w1,. . .,wδ be the roots of T1,. . .,Tδ, respectively, and let T be the full

δ

-ary tree rooted in w defined by the set of vertices V

(

T

)

₌V

(

T1

)

∪· · · ∪V

(

T_δ

)

∪

{

w

}

and edgesE

(

T

)

=E

(

T1

)

∪· · · ∪E

(

T_δ

)

∪

{

(

w,w1

)

,...,

(

w,w_δ

)

}

.LetS1,...,S_δ be outermultiset basesofT1,...,Tδ,respectively.Then

∀

i=j∈{1,...,δ}mTi

(

wi

|

Si

)

[



Ti

(

wi

)

]=mTj

(

wj

|

Sj

)

[



Tj

(

wj

)

]⇒

S1∪...∪S_δisanoutermultisetbasisofT.

Proof. Let S=S1∪...∪Sδ. Consider two vertices x and y in V(T)ࢨS such that dT

(

x,w

)

=dT

(

y,w

)

. We will prove that mT(x|S)=mT(y|S), which givesthat S is an outer multiset resolving set via application of Lemma5.1. Our proof is split intwocases,dependingonwhetherxandyarewithinthesamesub-branchornot.

sum of the leaf vertices that are not in S from all sub-branches, but T_i. Analogously, we obtain that mT

(

y

|

S

)

[



T

(

y

)

]= 

k∈{1,...,δ}\{j}mT_k

(

wk

|

Sk

)

[



T_k

(

wk

)

].Therefore,

mT

(

x

|

S

)

[



T

(

x

)

]− mT

(

y

|

S

)

[



T

(

y

)

]=mTj

(

wj

|

Sj

)

[



Tj

(

wj

)

]− mTi

(

wi

|

Si

)

[



Ti

(

wi

)

].

The premise is that

∀

i=j∈{1,...,δ}mT_i

(

wi

|

Si

)

[



T_i

(

wi

)

]=mTj

(

wj

|

Sj

)

[



T_j

(

wj

)

]. This gives mT_j

(

wj

|

S

)

[



T_j

(

wj

)

]− mT_i

(

wi

|

S

)

[



T_i

(

wi

)

]=0,whichleadstomT(x|S)[



T(x)]=mT(y|S)[



T(y)],whichimpliesthatmT(x|S)=mT(y|S).

Forthesecondcaseassumethatx_∈V(T_i)andy_∈V(T_i)forsomei_∈

{

1_,...,

δ}

.ThisimpliesthatmT(x|Si)=mT(y|Si),because

Siis an outer multisetresolving set inTi.Moreover, forevery vertexz∈V(T)ࢨV(Ti) it holdsthat dT

(

x,z

)

=dT

(

y,z

)

, which givestheexpectedresult:mT(x|S)=mT(y|S).

SofarwehaveprovedthatSisanouter multisetresolving set.ToprovethatSisabasis,we proceedbyshowingthat anyouter multisetresolving setS inT satisfiesthat thesets S∩V

(

T1

)

,...,S∩V

(

T_δ

)

are outer multisetresolving setsin T1,...,Tδ,respectively.GiventhatS1,...,Sδ areoutermultisetbases,thiswouldmeanthat Sisanoutermultisetresolving

setofminimumcardinality.

LetS₁₌S_∩V

(

T1

)

,...,S_δ=S∩V

(

T_δ

)

.AssumethatSi isnotanoutermultisetresolvingsetinTiforsomei∈

{

1,...,

δ}

. Thismeansthat thereexistvertices xandysuch that mT_i

(

x

|

Si

)

=mT_i

(

y

|

Si

)

.Because Tiis afull

δ

-arytree with

δ

>1and depth atleast two,there must exist two leaf vertices x1, x2 (resp.y1, y2) in Ti which are siblings andsatisfy d

(

x1,x

)

= d

(

x2,x

)

=



T_i

(

x

)

(resp.d

(

y1,y

)

=d

(

y2,y

)

=



T_i

(

y

)

).Consideringthat x1 andx2 (resp.y1 andy2) arefalse twins,weobtain

that x1∈S or x2∈S (resp.y1∈S or y2∈S). Thisimpliesthat x andy havethe same eccentricityinTi,because



T_i

(

x

)

<



T_i

(

y

)

⇒



T_i

(

y

)

∈mT_i

(

x

|

Si

)

,while



T_i

(

x

)

>



T_i

(

y

)

⇒



T_i

(

x

)

∈mT_i

(

y

|

Si

)

.GiventhatTiisafull

δ

-arytree,itfollowsthatxandy alsohavethesamedistancetotherootvertexw_i.Therefore,foreveryvertexz_∈V(T)_ࢨV(T_i)itholdsthat dT

(

x,z

)

=dT

(

y,z

)

, whichgivesmT

(

x

|

S

)

=mT

(

y

|

S

)

.ThiscontradictsthepremisethatSisamultisetresolvingset.HenceSiisanoutermultiset

resolvingset. 

Lemma5.2providesasufficientconditionforobtainingan outermultisetbasisofafull

δ

-arytreeTbyjoiningbasesof thefirstlevelbranchesofT.Thenextresultgoesfurther,byprovidingasufficientconditiontofinding

δ

/1outermultiset basesinT,basedon

δ

+1outermultisetbasesofthefirstlevelbranchesofT.Thisallowsustoexpressthesizeofanouter multisetbasisinarecurrenceequationand,consequently,provideaclosedformulafortheoutermultisetdimensionoffull

δ

-arytrees.

Theorem5.3. Let T_δ be a full

δ

-ary tree of depth. Letn be the smallest positiveinteger suchthat there exist

δ

+1 outer multisetbases S1,...,S_δ+1 in Tnδ satisfying that

∀

i=j∈{1,...,δ+1}m_Tδ

n

(

w

|

Si

)

[



Tnδ

(

w

)

]=mTnδ

(

w

|

Sj

)

[



Tnδ

(

w

)

], where w is the root of

T_nδ.Then,forevery≥ n,theoutermultisetdimensionofT_δ isgivenby

δ

−n× dimms

(

Tnδ

)

.

Proof.Weproceedbyinduction.

Hypothesis.Forsome≥ n,thefollowingtwoconditionshold:

1. There exists

δ

+1 outer multiset bases S1,...,S_δ+1 in Tδ satisfying that

∀

i=j∈{1,...,δ+1}mT_δ

(

w

|

Si

)

[



T_δ

(

w

)

]= m_Tδ



(

w

|

Sj

)

[



Tδ

(

w

)

]

2. TheoutermultisetdimensionofT_δ isgivenby

δ

−n× dimms

(

Tnδ

)

.

Clearly,thesetwoconditionsholdfor=n(basecase).The remainderofthisproof willbe dedicatedtofinding

δ

+1 outermultisetbasesR1,...,R_δ+1ofT_δ₊₁ thatsatisfy condition(1).Thesecond conditionwillfollowstraightforwardlyfrom

thesizeofthebasesR1,...,R_δ+1.

LetwbetherootofT_δ₊₁andwtherootofT_δ.Letw1,...,wδbethechildrenverticesofwinT_δ₊₁.Foreachsub-branch Tw_k of T_δ₊₁, withk∈

{

1,...,

δ}

, let

φ

k be an isomorphism fromT_δ to Tw_k.It follows that Sˆk=

{

φ

k

(

u

)

|

u∈Sk

}

is an outer multisetbasis of Tw_k,forevery k∈

{

1,...,

δ

+1

}

.Moreover, giventhat

∀

k∈{1,...,δ+1}mTw_k

(

wk

|

Sˆk

)

=mT_δ

(

w

|

Sk

)

, we conclude that

∀

i=j∈{1,...,δ+1}mTw_i

(

wi

|

Sˆi

)

[



Tw_i

(

wi

)

]=mTw_j

(

wj

|

Sˆj

)

[



Tw_j

(

wj

)

].

ByTheorem5.2,we obtainthat, forevery i∈

{

1,...,

δ

+1

}

, theset Ri=j∈{1,...,δ+1}\{i}Sˆj is anouter multiset basisof

T_δ₊₁.Moreover,foreveryi∈

{

1,...,

δ

+1

}

,thefollowingholds

m_Tδ +1

(

w 

_|

_R i

)

[



T_δ₊₁

(

w

)

]=  j∈{1,...,δ+1}\{i} mTwj

(

wj

|

Sˆi

)

[



Twj

(

wj

)

].

Fromtheequationaboveweobtainthatforeveryi_,j_∈

{

1_,...,

δ

₊1

}

_,

m_Tδ

Table 1

Three outer multiset bases of T 2

4 satisfying the premises of Theorem 5.3 . Vertices of T 2

4 have been labelled by using a breadth-first ascending order, starting by labelling the root node with 1 and finishing with the label 2 n+1 − 1 .

S 1 = { 22 , 24 , 14 , 25 , 26 , 16 , 28 , 18 , 2 , 8 , 30 , 20 , 21 } T ₄ 2 S 2 = { 22 , 12 , 24 , 14 , 26 , 16 , 28 , 18 , 6 , 8 , 30 , 20 , 21 } S 3 = { 22 , 24 , 14 , 25 , 26 , 16 , 17 , 28 , 18 , 8 , 30 , 20 , 21 }

Recall that

∀

k∈{1,...,δ+1}mTw_k

(

wk

|

Sˆk

)

=mT_δ

(

w

|

Sk

)

, which means that i= j⇒mTw_j

(

wj

|

Sˆj

)

[



Tw_j

(

wj

)

]=mTw_i

(

wi

|

Sˆi

)

[



Tw_i

(

wi

)

]. Therefore,weconcludethatT_δ₊₁andR1,...,R_δ+1 satisfythefirstconditionoftheinductionhypothesis,i.e.

∀

i=j∈{1,...,δ+1}mT_δ₊₁

(

w

|

Ri

)

[



T_δ₊₁

(

w

)

]=mT_δ₊₁

(

w

|

Rj

)

[



T_δ₊₁

(

w

)

].

Finally,observethat

|

R1

|

=

|

Sˆ2

|

× · · · ×

|

Sˆδ+1

|

=

δ

× dimms

(

T_δ

)

.Thesecondconditionoftheinductionhypothesis statesthat

dimms

(

T_δ

)

=

δ

−n× dimms

(

Tnδ

)

,whichgivesthatdimms

(

T_δ₊₁

)

=

δ

× dimms

(

T_δ

)

=

δ

+1−n× dimms

(

Tnδ

)

.  We end this section by addressing the problemof finding the smallest n such that T_nδ contains

δ

₊1 outer multiset bases S1,...,S_δ+1 satisfying the premisesof Theorem 5.3.We do so by developing a computer program1 that calculates

such number via exhaustivesearch. The pseudocode forthiscomputer program can be found in Algorithm1. Itreduces thesearchspacebyboundingthesizeofanoutermultisetbasiswiththehelpofLemma5.2(seeStep13ofAlgorithm1). Thatsaid,wecannotguaranteeterminationofAlgorithm1,essentiallyfortworeasons.First,thecomputationalcomplexity of each iterationof thealgorithm is exponential onthe size ofT_nδ while, atthe sametime, the size ofT_nδ exponentially increaseswithn.Second,thereisnotheoreticalguaranteesthatsuchanncanbefoundforevery

δ

.

Algorithm1 Givenanaturalnumber

δ

,findsthesmallestnsuchthatthefull

δ

-arytreeofdepthnsatisfiesthepremises ofTheorem5.3.

1: Letn=0andTnδafull

δ

-treeofdepthnrootedinw

2: min₌1 LowerboundonthecardinalityofabasisinT₁δ

3: max=

δ

− 1 UpperboundonthecardinalityofabasisinT₁δ

4: repeat

5: fori=mintomaxdo Eachoftheseiterationscanberaninparallel

6: LetBbeanemptyset 7: forallS⊆ V

(

Tnδ

)

s.t.

|

S

|

=ido

8: ifSisaresolvingsetthen

9: if

∀

_S∈BmT_nδ

(

w

|

S

)

[



T_nδ

(

w

)

]=mT_nδ

(

w

|

S

)

[



T_nδ

(

w

)

]then

10: B=B∪

{

S

}

11: ifB=∅then

12: break TheoutermultisetdimensionofTnδhasbeenfound

13: min=dimms

(

Tnδ

)

×

δ

SeeLemma5.2

14: max₌min₊

δ

_{− 1} Thisisthetrivialupperbound

15: n=n+1 16: until

|

B

|

≥

δ

+1 17: returnn

Despitethe exponential computationalcomplexity ofAlgorithm1, itterminatesfor

δ

=2. Inthis case,the smallestn

satisfyingthepremisesofTheorem5.3isn₌4.ThethreeoutermultisetbasesS1,S2 andS3 ofT42areillustratedinTable1

below.2 _{WerefertheinterestedreadertoAppendix6fora visualrepresentationofthebasesshowninTable1.Themain} corollaryofthisresultisthefollowing.

Corollary5.4. Theoutermultisetdimensionofafull2-arytreeT2

 ofdepthis: dimms

(

T_2

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1, if=1, 3, if=2, 6, if=3, 13, if=4, 2−4× 13, otherwise.

1 The computer program can be found at _{https://github.com/rolandotr/graph .}

Proof.The first fourcasesare calculated byan exhaustive searchusing acomputer program that canbe found athttps:

//github.com/rolandotr/graph.ThelastcasefollowsfromTheorem5.3. 

ItisworthremarkingthatAlgorithm1canbeparalellisedandhencebenefitfromacomputercluster.Runningthe algo-rithminahighperformancecomputingfacilityisthuspartoffuturework,whichmayleadtoterminationofAlgorithm1for valuesof

δ

higherthan2.

6. Conclusions

Inthispaperwehaveaddressedtheproblemofuniquely characterisingverticesinagraphbymeans oftheir multiset metricrepresentations.Wehavegeneralisedthetraditionalnotionofresolvabilityinsuch awaythat thenewformulation allows fordifferent structuralcharacterisations of vertices,including asparticular cases the onespreviously proposed in theliterature.We havepointedout afundamentallimitationaffectingpreviously proposedresolvabilityparameters based on the multiset representation,and have introduced a newnotion of resolvability, the outer multiset dimension, which effectivelyaddressesthislimitation.Additionally,wehaveconductedastudyofthenewparameter,wherewehaveanalysed itsgeneralbehaviour,determineditsexactvalueforseveralgraphfamilies,andproventheNP-hardnessofitscomputation, whileprovidinganalgorithmthatefficientlyhandlessomeparticularcases.

Acknowledgments

TheworkreportedinthispaperwaspartiallyfundedbyLuxembourg’sFondsNationaldelaRecherche(FNR),viagrants C15/IS/10428112(DIST) andC17/IS/11685812(PrivDA).

Appendix

Here,thereadercanfindgraphicalrepresentationsfordifferentoutermultisetbasesina full2-arytreeofdepth4.In thefigures,abasisisformedbythered-colouredvertices(Figs.5–7).

1

3

7

15

31

30

14

29

28

6

13

27

26

12

25

24

2

5

11

23

22

10

21

20

4

9

19

18

8

17

16

Fig. 5. The multiset representation of the root vertex with respect to the set of red-coloured vertices is {1, 2 2 , 4 10 }. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1

3

7

15

31

30

14

29

28

6

13

27

26

12

25

24

2

5

11

23

22

10

21

20

4

9

19

18

8

17

16