Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs

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Contents lists available atScienceDirect

Theoretical

Computer

Science

www.elsevier.com/locate/tcs

Induced

Subgraph

Isomorphism

on

proper

interval

and

bipartite

permutation

graphs

✩

Pinar Heggernes

a

,

Pim van ’t Hof

a

,

Daniel Meister

b

,

∗

,

Yngve Villanger

a

a_Department_of_Informatics,_University_of_Bergen,_P.O._Box_7803,₅₀₂₀_Bergen,_Norway b_Theoretical_Computer_Science,_University_of_Trier,_Germany

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received26January2012

Receivedinrevisedform6February2014 Accepted1October2014

Availableonline7October2014 CommunicatedbyG.Ausiello Keywords: Isomorphism Inducedsubgraphs Polynomial-time Fixed-parametertractable

Given twographs G and H asinput,the InducedSubgraphIsomorphism (ISI) problem istodecidewhetherG hasaninducedsubgraph thatisisomorphicto H.Thisproblem is NP-complete already when G and H are restricted to disjoint unions ofpaths, and consequentlyalso NP-complete onproper interval graphs and on bipartitepermutation graphs.We show that ISI can be solved in polynomial time onproper interval graphs andonbipartitepermutationgraphs,providedthat H isconnected.Asaconsequence,we obtainthatISIisﬁxed-parametertractableonthesetwographclasses,whenparametrised bythenumberofconnectedcomponentsof H.Ourresultscontrastandcomplementthe followingknown results:W[1]-hardnessofISI onintervalgraphs whenparametrisedby thenumber ofverticesof H, NP-completeness ofISI on connectedinterval graphs and on connected permutation graphs, and NP-completeness of SubgraphIsomorphism _on connectedproperintervalgraphsandconnectedbipartitepermutationgraphs.

1. Introduction

Westudythefollowingclassicaldecisionproblemforundirectedgraphs,whichhasapplicationsinavarietyofpractical areas[8,9].

InducedSubgraphIsomorphism_(ISI) Input: Twographs G andH .

Question: DoesG haveaninducedsubgraphisomorphictoH ?

Equivalently,thequestioniswhetherwecandeleteverticesandtheir incidentedgesfromG toobtainagraphisomorphic to H .ISIisageneralisationofseveralwell-knownproblemslike Clique, IndependentSet, LongestInducedPath,and Graph Isomorphism.As a consequenceofknownhardness results onsome oftheseproblems,ISIis NP-complete[9]aswell as W

[

1

]

-hardwhenparametrisedbythenumberofverticesinH [7].

✩ _This_work_is_supported_by_the_Research_Council_of_Norway_(NFR_FRITEK_grant_197548/V30)._A_preliminary_version_of_the_result_on_proper_interval graphswaspresentedatISAAC2010[11].

*

Correspondingauthor.

E-mailaddresses:pinar.heggernes@ii.uib.no(P. Heggernes),pim.vanthof@ii.uib.no(P. van ’t Hof),daniel.meister@uni-trier.de(D. Meister), yngve.villanger@ii.uib.no(Y. Villanger).

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Fig. 1. AnoverviewofthegraphclassesmentionedinthispaperandthecomplexitystatusofISIonthem.Here,“C→ D”istobeunderstoodas“Cisa subclassofD”.Bytheresultsofthispaper,ISIissolvableinpolynomialtimeonthegraphclasseswithlightgreyframes,whenH isconnected.Onthe graphclasseswithdarkgreyframes,itisknownfrompreviousworkthatISIisNP-completeevenwhenbothG andH areconnected.

Duetoitsimportance,ISIhasbeenstudiedonalargenumberofgraphclassesinpursuitofpolynomial-timesolvability; asummaryofsuchresultsisgiveninFig. 1.Unfortunately,theproblemturnsouttobenotoriouslydiﬃcult,andonlyvery fewnon-trivialpolynomial-timecasesareknown:ISIcanbe solvedinpolynomialtimewhenG isaforestandH isatree

[17],whenG and H are2-connectedouterplanargraphs[19],andwhenG isatriviallyperfectgraphandH isathreshold graph[1].Onthenegative side,ISIremainsNP-completewhen G isatreeandH isaforest[9],when G isacubicplanar graphandH isapath[9],andwhen G and H arebothconnectedcographs[5]orevenconnectedtrivially perfectgraphs

[1].Infact,ISIisNP-complete alreadywhen G and H aredisjointunions ofpaths [9,5].Thislast NP-completeness result directlyappliestoall hereditarygraphclassesthatcontainarbitrarily longinducedpaths.Inparticular,ISIisNP-complete whenG andH areproperintervalgraphsorwhenG and H arebipartitepermutationgraphs.

Inthispaper,weshowthatISIcanbesolvedinpolynomialtimewhentheinputgraphsareproperintervalgraphsand whenthey are bipartite permutationgraphs,provided that thesecond inputgraph H is connected.Incontrast,note that thesameisnottrueforintervalgraphsorforpermutationgraphs.Inparticular,evenifbothinputgraphsareconnected,ISI remainsNP-completeonintervalgraphs[16] andonpermutationgraphs[5].Anothercontrastisgivenbythefactthatthe relatedproblem SubgraphIsomorphism,whereoneasksfora subgraphratherthanan inducedsubgraph,isNP-complete whenthetwoinputgraphsarebothconnectedproperintervalgraphsorbothconnectedbipartitepermutationgraphs[14]. Toobtain ourpolynomial-timealgorithms, we ﬁrst solvethe followingproblem. Giventwo graphs G and H ,a vertex ordering

σ

G and a colouring fG for G,and avertex ordering

σ

H anda colouring fH for H , decidewhether thereis an isomorphismbetweenH andaninducedsubgraphofG thatpreservesthegivenvertexorderingsandmapstheverticesof H toverticesofG ofthesamecolour.Inthiscontext,preservingtheorderingsmeansthat theorderoftwoverticesofH accordingto

σ

H isthesameastheorderoftheir imagesinG accordingto

σ

G.We showthat if

σ

G and fG satisfy some speciﬁcconditions,thenwecandecideinpolynomialtimewhethersuchanisomorphismexists.Ourmainresultsarethen obtainedby showingthatproperintervalgraphsandbipartitepermutationgraphshavevertexorderingsthatsatisfy these conditions.

OurresultsextendtoshowingthatISIisﬁxed-parametertractableonproperintervalgraphsandonbipartitepermutation graphs,whentheparameteristhenumberofconnectedcomponentsofthesecond inputgraph H .Thiscomplementsthe resultsof[16],whereit isshownthatISIis W

[

1

]

-hard oninterval graphswhentheparameteris thenumberofvertices of H ,butitisﬁxed-parametertractableon intervalgraphswhentheparameteristhedifferencebetweenthenumbersof verticesofthetwoinputgraphs.TocompletethelistofknownparametrisedresultsonISI,let usmentionthatwhenthe parameteristhe numberofverticesof H ,ISIis ﬁxed-parametertractable whenbothinput graphsare planar[8]ortheir maximumvertexdegreeisboundedbyaconstant[3].

Ourpaperis organisedasfollows.Basic graph-theoretic deﬁnitionsandnotation are givenin Section 2.We solvethe above-mentionedproblemongraphswithcolouringsandvertexorderingsinSection3.Themainresultsonproperinterval graphs and bipartite permutation graphs are given in Sections 4 and 5, respectively. We end with the ﬁxed-parameter algorithmsinSection6andconcludingremarks.

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2. Deﬁnitionsandnotation

We considersimpleﬁniteundirectedgraphs.Agraph G isan orderedpair

(

V

,

E

)

,andV

=

V

(

G

)

isthevertexset ofG andE

=

E

(

G

)

istheedgeset ofG.EdgesofG aredenotedasuv,whereu andv areverticesofG.Ifuv isanedgeofG then u andv areadjacent andu isaneighbour ofv andv isaneighbour ofu inG.Otherwise,ifuv isnotanedgeofG,u and v arenon-adjacent inG.The(open)neighbourhood ofavertex u inG isthesetoftheneighboursofu inG,anditisdenoted as NG

(

u

)

.Theclosedneighbourhood ofu is NG[u

]

=

defNG

(

u

)

∪ {

u

}

.For X

⊆

V

(

G

)

,thesubgraphofG inducedby X , denoted asG

[

X

]

,isthegraphonvertexset X ,andforeveryvertexpair u

,

v from X ,uv isanedgeofG

[

X

]

ifandonlyifuv isan edgeofG.Agraph Gisaninducedsubgraph ofG ifthereexistsaset X

⊆

V

(

G

)

suchthat G

=

G

[

X

]

.Foravertex u ofG, we writeG

−

u to denotetheinducedsubgraph G

[

V

(

G

)

\ {

u

}]

ofG,thatis,G

−

u isthegraphobtainedfromG bydeleting vertex u.Aset X ofverticesofG isanindependentset of G iftheverticesin X arepairwisenon-adjacentinG.

Let G be a graph.Let k bean integer withk

≥

0 and let u

,

v be a vertex pairof G. A u

,

v-pathofG oflength k is a sequence

(

x0

,

. . . ,

xk

)

ofpairwisedifferentverticesofG suchthat x0

=

u andxk

=

v andxixi+1

∈

E

(

G

)

forevery 0

≤

i

<

k. Graph G isconnected if G hasau

,

v-pathforeveryvertexpair u

,

v ofG;otherwise,ifthereisavertexpair u

,

v ofG such that G hasnou

,

v-path,G iscalleddisconnected.Aconnectedcomponent of G isamaximalconnectedsubgraphofG.

Let G and H betwo graphs. We saythat H is isomorphicto G if thereis a bijective mapping

ϕ

from V

(

H

)

to V

(

G

)

such that foreveryvertexpair u

,

v of H ,uv

∈

E

(

H

)

ifandonlyif

ϕ

(

u

)

ϕ

(

v

)

∈

E

(

G

)

.Inthiscase, mapping

ϕ

is calledan isomorphism from H to G.In Section 3,we will decide the existence ofisomorphisms that satisfy additional conditions, namelythat are requiredtorespectvertexorderings andvertexcolours.Letk bean integerwithk

≥

1.Ak-colouring for G is amapping f that assigns a colourfrom theset

{

1

,

2

,

. . . ,

k

}

toeach vertexof G.Acolouring for G isa k-colouring for G forsome k.Acolouring f forG isproper ifforeveryedge uv ofG, f

(

u

)

=

f

(

v

)

.Avertexordering forG isalinear arrangement

σ

=

u1

,

. . . ,

unoftheverticesofG.Thereverse of

σ

isthevertexordering

un

,

. . . ,

u1

.Foravertexpair x

,

y ofG,wewritex

σ y ifx

=

uiandy

=

ujforsomeindices i

,

j andi

≤

j.If x

=

y andthusi

<

j,wewritex

≺

σ y.If x

≺

σ y, thenwesaythat x appearstotheleftof y intheordering

σ

,and y appearstotherightof x in

σ

.

Agraphclassishereditary ifitisclosedundertakinginduced subgraphs.Allgraphclassesmentionedinthispaperare hereditary[2,10].Furthermore,allgraphclassesinthispaperadmitwell-knowncharacterisationsthroughvertexorderings, andwe willoftendeﬁne graphclassesusingthesecharacterisations. Agraph G is acomparabilitygraph ifit hasavertex ordering

σ

such that for every vertextriple u

,

v

,

w of G with u

≺

σ v

≺

σ w , uv

∈

E

(

G

)

and v w

∈

E

(

G

)

implies u w

∈

E

(

G

)

[10].Suchvertexorderingsarecalledcomparabilityorderings.Agraphisacocomparabilitygraph ifitscomplementisa comparabilitygraph.Equivalently,agraph G isacocomparabilitygraphifandonlyifithasavertexordering

σ

suchthatfor everyvertextriple u

,

v

,

w ofG withu

≺

σ v

≺

σ w ,u w

∈

E

(

G

)

impliesuv

∈

E

(

G

)

orv w

∈

E

(

G

)

[13].Suchvertexorderings arecalledcocomparabilityorderings.Agraphisbipartite ifithasaproper2-colouring,i.e.,ifitsvertexsetcanbepartitioned into two independent sets. Bipartite graphs are comparability graphs. Proper interval graphs andbipartite permutation graphs are both subclassesof cocomparabilitygraphs [2,10]; seealso Fig. 1. The deﬁnitionsor characterisations, through vertexorderings, andnecessarypropertiesof properinterval graphsandofbipartite permutation graphswillbe givenat thebeginnings ofSections4and5.Comprehensivesurveysonpropertiesoftheconsidered graphclassescanbe foundin monographssuchas[2]and[10].

Finally, forabriefbackgroundon parametrisedcomplexity, aparametrisedproblem hasapartofits input, typicallyan integer,identiﬁedastheparameter.Aparametrisedproblemisﬁxed-parametertractable ifthereisanalgorithmforsolvingit intime f

(

k

)

·

nO(1),wheren isthetotalinputsize,k istheparameter,and f isafunctionthatdependsonlyonk andthat doesnot involven.There isahierarchyofintractableparametrised problemclasses[7].Fortheresultsmentionedinthis paper,itissuﬃcienttonoticethataparametrisedproblemisunlikelytobeﬁxed-parametertractableifitisW

[

1

]

-hard.

3. ISIongraphswithcolouringsandvertexorderings

LetG and H bearbitrarygraphsandlet

σ

beavertexorderingforG.AssumethatG hasaninducedsubgraph Gsuch that H isisomorphictoG.Then,thereareavertexordering

τ

forH andatotalinjectivemapping

ϕ

:

V

(

H

)

→

V

(

G

)

such thatthefollowingtwoconditionsaresatisﬁedforeveryorderedvertexpair u

,

v of H :

1) uv

∈

E

(

H

)

ifandonlyif

ϕ

(

u

)

ϕ

(

v

)

∈

E

(

G

)

2) u

≺

τ v ifandonlyif

ϕ

(

u

)

≺

σ

ϕ

(

v

)

.

The ﬁrstcondition isthe isomorphismconditionfor H and G.Forthesecond condition, observethat

τ

can beobtained from

σ

byrestrictiontotheverticesofGandapplying

ϕ

−1_._In_this_section,_we_will_consider_this_type _of_an_isomorphism problem.

Themainresultofthissectionisaneﬃcientalgorithmthatdecidestheexistenceofanisomorphismthatrespectsgiven colouringsandvertexorderings.LetG and H be twographs,andlet

σ

and

τ

bevertexorderings forrespectivelyG and H .Let

ϕ

:

V

(

H

)

→

V

(

G

)

beatotalmapping.Ifforevery orderedvertexpair u

,

v of H ,u

≺

τ v implies

ϕ

(

u

)

≺

σ

ϕ

(

v

)

then wesaythat

ϕ

is

(

σ

,

τ

)

-monotone.Observethata

(

σ

,

τ

)

-monotonemappingisinjective.Assumethat H isisomorphictoG, andlet

ϕ

:

V

(

H

)

→

V

(

G

)

beanisomorphismfromH toG.If

ϕ

is

(

σ

,

τ

)

-monotone,wesaythat

ϕ

isa

(

σ

,

τ

)

-isomorphism andthat H is

(

σ

,

τ

)

-isomorphic to G.Wesaythat H is

(

σ

,

τ

)

-isomorphictoaninducedsubgraph ofG ifthereisaninduced

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Fig. 2. Depictedisavertexorderingforagraph.Theverticesofthegrapharecolouredwiththecolours “triangle”and“circle”.Weiterativelyapplythe predecessorfunction pred,startingattherightmostvertexofcolour “triangle”,anditlistsallverticeswithcolour “triangle”,intheorderofappearancein thevertexordering,fromrighttoleft.

subgraph G of G such that H is

(

σ

,

τ

)

-isomorphic to G, where

σ

is the restriction of

σ

to the vertices of G. Our algorithmic resultofthissection aimsatdecidingforgivengraphs G and H andvertexorderings

σ

and

τ

whetherH is

(

σ

,

τ

)

-isomorphictoaninducedsubgraphofG.

In additionto the vertexordering monotonicity ofthe desiredisomorphism, we alsoask for isomorphisms that map betweenequalclassesofvertices.Weformalisethisclassnotionbycolours.LetG and H betwographsandlet fG and fH becolouringsforrespectivelyG andH .Let

ϕ

:

V

(

H

)

→

V

(

G

)

beanarbitrarytotalmapping.Wecall

ϕ

colour-preserving for

(

G

,

fG

)

and

(

H

,

fH

)

,if fH

(

x

)

=

fG

(

ϕ

(

x

))

foreveryvertex x ofH .Ouralgorithmwilldecidetheexistenceofcolour-preserving isomorphisms.

Beforeweformallypresentouralgorithm,wegiveaninformaldescription.Thealgorithmreceivesasinputtwographs G and H , as well as two colourings fG and fH and two vertex orderings

σ

and

τ

for respectively G and H . The algo-rithmdecideswhetherH iscolour-preserving

(

σ

,

τ

)

-isomorphictoaninducedsubgraphofG,whichmeansthatthereisa

(

σ

,

τ

)

-isomorphismfromH toan inducedsubgraph ofG thatisalsocolour-preserving.Fortheinitialstep,thealgorithm mapstheverticesofH totherightmostpossibleverticesofG,onlyrespectingthecolouringsandvertexorderings.Wecan saythat the algorithmbegins withthe rightmostcolour-preserving

(

σ

,

τ

)

-monotonemapping. Now,inrounds, the algo-rithmcheckswhetherthecurrentcolour-preserving

(

σ

,

τ

)

-monotonemappingisalsoanisomorphism.Ifyes,thealgorithm hasfound acolour-preserving

(

σ

,

τ

)

-monotoneisomorphismfrom H to an induced subgraphof G.Otherwise,there isa “conﬂicting” vertexpair u

,

v of H that violatesthe isomorphismcondition,which meansthat u andv are adjacentin H whiletheirimagesarenon-adjacentinG,orvice-versa.Basedonthesetwopossiblecases,thealgorithmcomputesanew colour-preserving

(

σ

,

τ

)

-monotonemappingbypushingu orv furtherleft,andentersanewround.

Thecrucialauxiliary operationis thepushoperationfora vertex.Intuitively,thepushoperation appliedtoa vertex u ﬁndstheclosestvertextotheleftofu ofaspeciﬁedcolour,ifavertexofthatcolourexists.Webasethepushoperationon the“predecessor”function.LetG beagraph,let f beacolouringforG andlet

σ

=

x1

,

. . . ,

xnbeavertexorderingforG. Leta beavertexpositionindexwith1

≤

a

≤

n andletc beacolour.Then,

pred₍_σ_,_f₎

(

a

,

c

)

=

defmax

a

:

1

≤

a

<

a and f

(

xa

)

=

c

∪ {

0

}

.

Inwords,whengivenasinputavertexposition a andacolour c,thepredecessorfunction pred₍_σ_,f)considersallvertices

totheleftofxa invertexordering

σ

thathavecolour c anditoutputsthelargestsuchposition,unlessnovertextotheleft ofxain

σ

hascolour c,inwhichcaseitoutputs 0.Notethatxaitselfdoesnotneedtohavecolour c.Asanexampleforthe iteratedapplicationofpred toagraphwitha2-colouring,Fig. 2indicatesthepredecessorofeachtriangle-colouredvertex. We arenow readytopresentthealgorithm. Itis calledOrderedInducedSubgraph, andwe oftenuseOISforshort.It receivesasinputa6-tuple

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

ofthefollowingcomponents:

•

G isagraph, fG isacolouringforG,and

σ

isavertexorderingforG

•

H isagraph, fH isacolouringforH ,and

τ

isavertexorderingof H .

ThealgorithmdecideswhetherH iscolour-preserving

(

σ

,

τ

)

-isomorphictoaninducedsubgraphofG. Algorithm OrderedInducedSubgraph (OIS)

Input A graph G with colouring fG and vertex ordering

σ

=

x1

, . . . ,

xn; a graph H with colouring fH and vertex ordering

τ

=

y1

, . . . ,

yr. begin

if fG

(

xi

)

=

fH

(

yr

)

for every 1

≤

i

≤

n then reject end if;

let arbe the largest vertex position index such that fG

(

xar

)

=

fH

(

yr

)

;

for i

=

r

−

1 downto 1 do let ai

=

pred(σ,fG)

(

ai+1

,

fH

(

yi

))

end for;

if a1

=

0 then reject end if;

while H is not

(

σ

,

τ

)

-isomorphic to G

[{

xa1

, . . . ,

xar

}]

do

let i,j be a vertex position index pair with 1

≤

i

<

j

≤

r such that yiyj

∈

/

E

(

H

)

if and only if xaixaj

∈

E

(

G

)

;

if yiyj

∈

/

E

(

H

)

then push

₍

i

₎

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push

₍

_j

₎

end if end while;

return

(

a1

, . . . ,

ar

)

and accept

end. Subroutine push(b

)

begin set ab

=

pred(σ,fG)

(

ab

,

fH

(

yb

))

; while ( ab

≤

ab−1 and b

≥

2 ) do set ab−1

=

pred(σ,fG)

(

ab

,

fH

(

yb−1

))

; set b

=

b

−

1 end while;

if a1

=

0 then reject end if end.

The algorithm consists of a main procedure and a subroutine called push. The main procedure iteratively computes a sequence

a

0

_,

_{. . . ,}

_a

h

_of_vertex_position_index_tuples,_and_each_such _tuple_is_of_the_form

₍

_a

1

,

. . . ,

ar

)

.Noteherethat input graph H hasr vertices.The for loopofthemainprocedurecomputestheinitialisingtuple

a

0,andthe while loopcomputes the rest of the tuples

a

1

,

. . . ,

a

h. Note that only tuple

a

h is output, andthis is done only if the algorithm accepts. An execution of the while loop body of the main procedure is called a round. For an integer e

≥

0, the index tuple

a

e

=

(

ae₁

,

. . . ,

aer

)

istheresultofround e,andthevaluesofae1

,

. . . ,

aer arethevaluesofrespectivelya1

,

. . . ,

ar ofthe algorithm attheendofround e,inparticular,aftertheapplicationofSubroutine push.Theexecutionofthe for loopisequivalentto round 0.Fortwo r-tuples

b

= (

b1

,

. . . ,

br

)

and

c

= (

c1

,

. . . ,

cr

)

,we write

b

≤ c

ifbi

≤

ci forevery1

≤

i

≤

r, andwewrite

b

< c

if

b

≤ c

andbi

<

ciforsome1

≤

i

≤

r.

We now analysethecorrectness ofthealgorithm. Wedoour analysisintwo steps:ﬁrst,we considerthe correctness ofan acceptdecision, andsecond, weconsiderthecorrectnessofarejectdecision. Wewillseethatan acceptdecisionis always correct(Lemma 3.1), andwe will see that a rejectdecision iscorrect incasethe input satisﬁessome conditions (Lemma 3.3).

Lemma3.1.LetG beagraphwithcolouring fGandvertexordering

σ

=

x1

,

. . . ,

xn,andletH beagraphwithcolouring fH and vertexordering

τ

=

y1

,

. . . ,

yr.WhenAlgorithm OIS isrunoninput

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

,thefollowingholds:

1) forevery0

≤

e

≤

h,where

a

e

= (

ae₁

,

. . . ,

ae

r

)

: ifae1

≥

1 thena e

1

<

· · · <

aer 2) forevery0

≤

e

≤

h,where

a

e

= (

ae₁

,

. . . ,

ae_r

)

: ifae₁

≥

1 thenfH

(

yi

)

=

fG

(

xae

i

)

forevery1

≤

i

≤

r 3) forevery1

≤

e

≤

h:

a

e

< a

e−1

4) if OIS accepts on input

₍

G

_,

f_G

_,

σ

;

H

_,

f_H

_,

τ

₎

and outputs

₍

a₁

_,

_{. . . ,}

a_r

₎

, then H is colour-preserving

₍

σ

_,

τ

₎

-isomorphic to G

[{

xa1

,

. . . ,

xar

}]

.

Proof. Let

a

0

,

. . . ,

a

h

bethecomputedvertexpositionindextuplesequence.

We prove statements 1 and2 simultaneously, by induction on e. Lete

=

0, andassume that a0

1

≥

1.Then, fH

(

yr

)

=

fG

(

x_a0

r

)

duetothechoice ofa

0

r,anda0i

<

a 0

i+1 and fG

(

xa0_i

)

=

fH

(

yi

)

forevery 1

≤

i

<

r, accordingtothedeﬁnitionofthe predecessorfunction pred.Thus,

a

0 satisﬁesthetwoconditions.

Now, let e

≥

1, assume that ae₁

≥

1, and assume that

a

e−1 satisﬁes the two conditions. The new index tuple

a

e is computed during an executionof the while loop body, thus, it is the resultof an execution ofSubroutine push. Since ae₁−1

<

· · · <

ae_r−1byassumption,theiteratedapplicationofthepredecessorfunctioninSubroutine push yieldsae₁

<

· · · <

ae_r. Itiscrucialtorecallourassumptionaboutae₁

≥

1 here.Thesameassumptionalsoimplies fG

(

xae

i

)

=

fG

(

xaei−1

)

=

fH

(

yi

)

for every 1

≤

i

≤

r, whichis duetothedeﬁnition ofpred andourassumption about

a

e−1.Thus,

a

e indeedsatisﬁesthetwo conditions.

We prove statement 3.Let1

≤

e

≤

h, andweconsider

a

e−1 and

a

e. Since

a

e isobtainedfrom

a

e−1 by an application of Subroutine push,

a

e−1

≤ a

e is a directconsequence ofpush andthe propertiesof thepredecessor function pred, and

a

e−1

= a

e followsfromtheﬁrstassignmentinpush_.

Weprovestatement 4.AssumethatOISacceptsoninput

₍

G

_,

f_G

_,

σ

;

H

_,

f_H

_,

τ

₎

.Then,a0

1

≥

1,andround h isthelastfully executed round,andah

1

≥

1. So, H is

(

σ

,

τ

)

-isomorphic to G

[{

xah

1

,

. . . ,

xahr

}]

,andaccordingto the resultsof statements 1 and 2, H iscolour-preserving

(

σ

,

τ

)

-isomorphictoG

[{

x_ah

1

,

. . . ,

xahr

}]

.

2

(6)

Fig. 3. AnillustrationoftheleftandrightumbrellaconditionsofDeﬁnition 3.2.Thevertices u,v,w satisfyu≺σv≺σw,andwehaveverticesoftwo

colours,namelyblackandwhite.

Forthesecondsteptowardacorrectnessproofforthealgorithm,weconsidertherejectionbehaviourofthealgorithm. Beforeweproceedwiththisstep,wemakesomegeneralobservations.LetG andH betwographs,andlet

σ

and

τ

be ver-texorderingsforrespectivelyG and H .ObservethatitcanbedecidedinpolynomialtimewhetherH is

(

σ

,

τ

)

-isomorphic to G,since only one mappingfrom V

(

H

)

to V

(

G

)

is possible.On theother hand,when generalisingthe problemto in-ducedsubgraphs,decidingwhetherH is

(

σ

,

τ

)

-isomorphictoaninducedsubgraphofG isNP-completeongeneralgraphs. A straightforward reduction canbe constructedfrom Clique. And to fullysatisfy the setting ofour specialisomorphism problem, we can additionally equip G and H with a 1-colouring.As a consequence, we cannot expect ouralgorithm to workcorrectlyonallpossibleinputs.Werestrictourinputsandrequirethemtosatisfytwospecialconditions.Thesetwo conditionsaredeﬁnednext.

Deﬁnition3.2.LetG beagraph,let f beacolouringforG,andlet

σ

beavertexorderingfor G.

1)

(

G

,

f

,

σ

)

satisﬁestheleftumbrellacondition ifforeveryvertextriple u

,

v

,

w ofG withu

≺

σ v

≺

σ w and f

(

v

)

=

f

(

w

)

, u w

∈

E

(

G

)

impliesuv

∈

E

(

G

)

.

2)

(

G

,

f

,

σ

)

satisﬁestherightumbrellacondition ifforeveryvertextripleu

,

v

,

w ofG withu

≺

σ v

≺

σ w and f

(

u

)

=

f

(

v

)

, u w

∈

E

(

G

)

impliesv w

∈

E

(

G

)

.

Observethatthetwoumbrellaconditionsaresymmetric,whichmeansthat

(

G

,

f

,

σ

)

satisﬁestheleftumbrellacondition ifandonlyif

(

G

,

f

,

σ

R

₎

_satisﬁes_the_right_umbrella_condition,_where

_σ

R _is_the_reverse_of

_σ

_._An_illustrating_description_of thetwoumbrellaconditionsisgiveninFig. 3.

Lemma3.3.LetG beagraphwithcolouring fG andvertexordering

σ

=

x1

,

. . . ,

xnsuchthat

(

G

,

fG

,

σ

)

satisﬁestheleftand rightumbrellaconditions,andlet H beagraphwithcolouring fH andvertexordering

τ

=

y1

,

. . . ,

yr.IfAlgorithm OIS rejects input

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

,thenH isnotcolour-preserving

(

σ

,

τ

)

-isomorphictoanyinducedsubgraphofG.

Proof. Recall thedeﬁnitionsprecedingLemma 3.1 andthe technicalresultsfromLemma 3.1;we willmake heavyuse of these throughoutthe proof. Let

a

0

,

. . . ,

a

h

be the computedindex tuplesequence. We prove the contraposition ofthe statementofthelemma.Let

σ

=

x1

,

. . . ,

xnand

τ

=

y1

,

. . . ,

yr.

WeassumethatH iscolour-preserving

(

σ

,

τ

)

-isomorphictoaninducedsubgraphofG.Then,thereareindices d1

,

. . . ,

dr with1

≤

d1

<

· · · <

dr

≤

n suchthatH iscolour-preserving

(

σ

,

τ

)

-isomorphictoG

[{

xd1

,

. . . ,

xdr

}]

.Recallthatthisparticularly means fH

(

yi

)

=

fG

(

xdi

)

forevery1

≤

i

≤

r.Welet

d

=

def

(

d1

,

. . . ,

dr

)

.Itwillbeimportantlaterthat

di

≤

pred(σ,fG)

di+1

,

fH

(

yi

)

istrueforevery1

≤

i

<

r.We showthatOIS_accepts_with_output_tuple

_a

h _and_that

_d

≤ a

h _holds._To_prove_the _result,_we showbyinductionone that

d

≤ a

e holds.

Inductionbase

Recallthechoiceof

d

andthedeﬁnitionof

a

0,andobservethatthefollowingholds:

fG

(

xdr

)

=

fG

(

xa0

r

)

and dr

≤

a

0 r

.

Weconsidertheothercomponentsof

d

and

a

0.Let1

≤

i

<

r.RecallfromtheinitialisingstepofOIS:

a0_i

=

pred₍_σ_,fG)

a0_i₊₁

,

fH

(

yi

)

.

Itfollowsthata0_i

<

a_i0₊₁and fG

(

xj

)

=

fH

(

yi

)

,foreverya0i

<

j

<

a 0

i+1.Sincedi

<

di+1

≤

a0i+1and fG

(

xdi

)

=

fH

(

yi

)

,itfollows thatdi

≤

a0_i isthecase.Weconclude

d

≤ a

0,whichprovestheinductionbase.

Inductionstep

We consideran arbitraryround e, for0

≤

e

<

h.We assume

d

≤ a

e,andwe show

d

≤ a

e+1.Suppose fora contradiction that

d

≤ a

e+1.Then,thereexists anindex t with1

≤

t

≤

r suchthat ae_t+1

<

dt.Due totheassumption

d

≤ a

e,thismeans ae_t+1

<

dt

≤

aet.

(7)

Wemakesomepreparations.Recallthat

a

e+1isobtainedfrom

a

e byanapplicationofSubroutine push.Letp

,

q bethe conﬂictingindexpairchosen atthebeginningofround e

+

1,andwecanassume p

<

q.Accordingtothechoiceofp and q,itholds: ypyq

∈

/

E

(

H

)

ifandonlyif xae

pxaeq

∈

E

(

G

)

.Recallthatpush isinvokedwith p or q.Letb bethenumberpush isinvokedwith.The followingisa simpleobservationfromthedeﬁnitionofpush_:_ae+1

i

=

aei foreveryb

<

i

≤

r. Asanas simpleasimportantconsequence:t

≤

b.Furthermore,accordingtopush:

ae_i₋+₁1

=

pred₍_σ_,_f_G₎

ae_i+1

,

fH

(

yi−1

)

foreveryt

<

i

≤

b.

We consider the valuesof dt

,

. . . ,

db and aet+1

,

. . . ,

a e+1

b . Let i be an indexwitht

<

i

≤

b,andassume di

≤

a e+1 i . The monotonicityofthepredecessorfunction pred₍_σ_,_f_G₎shows:

pred₍_σ_,fG)

di

,

fH

(

yi−1

)

≤

pred₍_σ_,fG)

ae_i+1

,

fH

(

yi−1

)

,

whichimplies di−1

≤

pred(σ,fG)

di

,

fH

(

yi−1

)

≤

pred(σ,fG)

ae_i+1

,

fH

(

yi−1

)

=

ae_i₋+₁1

,

sothatdi

≤

aei+1 impliesdi−1

≤

aei−+11.Fromthis,weconcludethatdb

≤

ae_b+1 impliesdt

≤

ate+1.Sinceate+1

<

dt accordingto ourassumptions,wecanconcludeae_b+1

<

db,andthus,aeb+1

<

db

≤

aeb.Furthermore,since

fG

(

x_ae+1 b

)

=

fG

(

xdb

)

=

fG

(

xa e b

)

=

fH

(

yb

)

and a e+1 b

=

pred(σ,fG)

ae_b

,

fH

(

yb

)

,

weconcludedb

=

aeb.Informally,thismeansthatxae

b isacorrectchoiceforyb.

Weemploydb

=

abe toconstructthedesiredcontradiction.Wedistinguishbetweenthetwo casesaboutthevalue ofb: eitherb

=

p orb

=

q.

Case1.pushisinvokedwithparametervalue b

=

p.

Proofofthecase. Wehavethefollowingsituation:

ypyq

∈

/

E

(

H

)

;

xdpxdq

∈

/

E

(

G

)

;

xaepxaeq

∈

E

(

G

)

;

dp

=

a

e

p

;

p

=

b

<

q

.

Recallfromtheabovethatb

=

p anddb

=

aebimpliesdp

=

aep.Sinceaep

=

dp

<

dq

≤

aeqandxdpxdq

∈

/

E

(

G

)

andxdpxaeq

∈

E

(

G

)

, itclearlyfollowsdq

=

aeq,sothataep

=

dp

<

dq

<

aeq musthold.

Weconstructthedesiredcontradiction.Observethat fG

(

xdq

)

=

fG

(

xaeq

)

.Sincexaepxaeq

∈

E

(

G

)

andsince

(

G

,

fG

,

σ

)

satisﬁes theleftumbrellacondition,itfollowsthat xdpxdq

∈

E

(

G

)

musthold,contradictingtheobservedsituation.

2

Case2.pushisinvokedwithparametervalue b

=

q.

Proofofthecase. Wehavethefollowingsituation:

ypyq

∈

E

(

H

);

xdpxdq

∈

E

(

G

);

xaepxaeq

∈

/

E

(

G

);

dq

=

a

e

q

;

p

<

q

=

b

.

Recallfromtheabovethatb

=

q anddb

=

aebimpliesdq

=

a

e

q.Sincedp

≤

aep

<

aeq

=

dqandxdpxdq

∈

E

(

G

)

andxapexdq

∈

/

E

(

G

)

, itclearlyfollowsdp

=

aep,sothatdp

<

aep

<

aqe

=

dqmusthold.

Since xaepxaeq

∈

/

E

(

G

)

and fG

(

xdp

)

=

fG

(

xaep

)

and since

(

G

,

fG

,

σ

)

satisﬁes the rightumbrella condition, it followsthat

xdpxdq

∈

/

E

(

G

)

,whichgivesthedesiredcontradiction.

2

Wesummarisetheshownresults.Wehaveproved

d

≤ a

h,and

a

h

<

· · · < a

0 isduetostatement 3ofLemma 3.1.So,OIS doesnotreject,whichmeansthatOISaccepts.

2

Corollary3.4.LetG beagraphwithcolouring fG andvertexordering

σ

=

x1

,

. . . ,

xnsuchthat

(

G

,

fG

,

σ

)

satisﬁestheleftand right umbrellaconditions,andlet H beagraph withcolouring fH andvertexordering

τ

=

y1

,

. . . ,

yr.Algorithm OIS on in-put

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

decidesinpolynomialtimewhetherH iscolour-preserving

(

σ

,

τ

)

-isomorphictoaninducedsubgraphof G.

Proof. Thecorrectnessofthealgorithmisthejoinedresultofstatement 4ofLemma 3.1andofLemma 3.3.

We consider therunning timeof thealgorithm. Eachround ofthealgorithm haspolynomial running time,andsince

a

h

<

· · · < a

1

< a

0forthecomputedsequence

a

0

,

. . . ,

a

h

ofvertexpositionindextuplesduetostatement 3ofLemma 3.1, itremainstoboundparameter h.Thevectorsumsaredecreasingwithincreasingindexanda0

1

+ · · · +

a0r

≤

r

·

n,sothatwe concludeh

≤

n2 _as_an_upper_bound_on_the_number_of_executed_rounds.

₂

(8)

ondglance,suchrestrictionsareinavoidable,aswealreadydiscussedinthissection.WewillapplyOIStosolvetheinduced subgraph isomorphismproblemonproper intervalgraphs andbipartite permutationgraphs. Andfor thesegraphclasses, wecanchooseinputsthatinfactsatisfythetwoconditionsnaturally.

WeconsiderOISmoreclosely.IfOISacceptsthenitalsooutputsavertexpositionindextuple,namely

a

h,whichdeﬁnes a colour-preservingmonotone induced subgraphisomorphism,accordingtostatement 4ofLemma 3.1.Let

d

be an index tuplethatdeﬁnesanarbitrarycolour-preservingmonotoneinducedsubgraphisomorphism.TheproofofLemma 3.3shows that

d

≤ a

hholds.Wecanthereforesaythattheoutputindextupledeﬁnesthe“maximum”or“rightmost”colour-preserving monotoneinducedsubgraphisomorphism.Thisisausefulobservation,usefulwhenaskingforisomorphismsofveryspecial properties.

Another interesting observation about OIS concerns the chosen conﬂicting index pair i

_,

j. The algorithm chooses an arbitrarysuchindexpairamongallpossibleconﬂictingindexpairs,andtheresultofthealgorithm,acceptanceorrejection as well as the output index tuple, is not inﬂuenced by the actual choice. This is noteworthy for theoretical as well as implementation aspects.

Finally, consider the related subgraph isomorphism problem: given two graphs G and H , decide whether H is iso-morphic to a (notnecessarily induced) subgraph G of G. We can modify OIS to decide this problem: conﬂicting index pairs are those i

,

j for which yiyj

∈

E

(

H

)

and xaixaj

∈

/

E

(

G

)

. It is an easy exercise to modify the proofs ofthis section to show thefollowing result:thereis apolynomial-time algorithmthat decides on input

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

whether H iscolour-preserving

(

σ

,

τ

)

-isomorphictoasubgraph ofG,assumingthat

(

G

,

fG

,

σ

)

satisﬁestherightumbrellacondition. If G isan interval graph, fG is a 1-colouring forG and

σ

is an interval ordering for G,thisis the case. Neglectingthe

(

σ

,

τ

)

-monotonicity, the resulting SubgraphIsomorphismproblem is NP-complete already on connected proper interval graphs[14].

4. ISIonproperintervalgraphs

Properintervalgraphsarecocomparabilitygraphsofspecialproperties,andtheyhaveacharacterisationthroughvertex orderings.Let G beagraph.Avertexordering

σ

forG iscalledaproperintervalordering ifforeveryvertextriple u

,

v

,

w ofG with u

≺

σ v

≺

σ w , u w

∈

E

(

G

)

impliesuv

∈

E

(

G

)

and v w

∈

E

(

G

)

.Agraphisaproperintervalgraph ifithasaproper interval ordering[15].We onlyconsiderproper interval graphsthrough thisvertexorderingcharacterisation. Observefor every vertexordering

σ

for G:

σ

is aproper interval orderingfor G if andonlyifthe reverse of

σ

is a properinterval orderingfor G.Alsoobservethat everyproper intervalorderingis acocomparabilityordering.Itcan bedecided inlinear time whether a given graphis a proper interval graph,and ifso, a proper interval ordering can be generated in linear time[15].

First,weprovethatproperinterval orderingsareuniqueuptoreversalanduptosetsofverticeswiththesame neigh-bourhoods.Tobe moreprecise,let G beaconnectedproper intervalgraphandlet

σ

and

τ

be properinterval orderings for G. Assume that

σ

=

x1

,

. . . ,

xn and

τ

=

y1

,

. . . ,

yn. We say that

σ

and

τ

are stronglyneighbourhood-equivalent if NG[xi]

=

NG[yi]forevery1

≤

i

≤

n.Wewillshowthat

σ

isstronglyneighbourhood-equivalentto

τ

ortothereverseof

τ

. InordertoprovethisstatementasProposition 4.2below,itsuﬃcestoprovethenextlemma,whosetechnicalstatements willbereferred toseparately laterinthepaper.The ﬁrststatement ofLemma 4.1showsthat iftheleftmostverticesof

σ

and

τ

havethesameclosedneighbourhoodthen

σ

and

τ

arestronglyneighbourhood-equivalent.Thesecondstatementof thelemmashowsthattheclosedneighbourhoodsoftheﬁrstverticesof

σ

and

τ

areequalortheclosedneighbourhoods oftheﬁrstvertexof

σ

andthelastvertexof

τ

areequal.AlthoughLemma 4.1hasbeenusedimplicitlyinseveralprevious works[4,6,12,15],ithasnotbeenstatedasaresultonitsownandprovedseparatelybefore.

Lemma4.1.LetG beaconnectedproperintervalgraph.Let

σ

=

x1

,

. . . ,

xnand

τ

=

y1

,

. . . ,

ynbeproperintervalorderingsforG. 1) IfNG

[

x1

]

=

NG

[

y1

]

,then

σ

and

τ

arestronglyneighbourhood-equivalent.

2) Ifx1

≺

τ xn,thenNG[x1

]

=

NG[y1

]

,and ifxn

≺

τ x1,thenNG[x1

]

=

NG[yn

]

.

Proof. Since the two statements trivially hold for the casen

=

1, we assume n

≥

2. The following properties ofproper interval orderings willbe importantfortheproof. Consider

σ

. Since G isconnected, xixi+1

∈

E

(

G

)

forevery 1

≤

i

<

n.It directly follows that

(

x1

,

. . . ,

xn

)

is an x1

,

xn-pathof G.Particularly note that x2

∈

NG

[

x1

]

. Furthermore,for every vertex triple u

,

v

,

w ofG withu

=

x1 andu

≺

σ v

σ w and w

∈

NG

(

u

)

,itholdsthat NG

[

u

]

⊆

NG[v

]

⊆

NG[w

]

.Since

τ

isalsoa properintervalorderingforG,thesamepropertiesanalogouslyholdfor

τ

.

Proofof1)

Weassume NG[x1

]

=

NG[y1

]

.Ifx2

=

y2then NG[x2

]

=

NG[y2

]

,ifx2

=

y1then y1

≺

τ y2

τ x1,andthus,NG[x1

]

⊆

NG[x2

]

=

NG[y1

]

⊆

NG[y2

]

⊆

NG

[

x1

]

,if y2

=

x1 then,analogously, NG[y2

]

=

NG[x2

]

, andif x1

=

y2 and x2

=

y2 andx2

=

y1 then x1

≺

σ x2

≺

σ y2 andy1

≺

τ y2

≺

τ x2 andNG[x2

]

⊆

NG[y2

]

andNG[y2

]

⊆

NG[x2

]

.Thus,NG

[

x2

]

=

NG

[

y2

]

inallcases.

(9)

Let j besuchthat yj

=

x1;notethat NG[y1

]

=

NG

[

x1

]

=

NG

[

yj].If j

=

1 thenx1

=

y1,and

x2

,

. . . ,

xnand

y2

,

. . . ,

yn are properinterval orderings for G

−

x1. If j

≥

2 then

x2

,

. . . ,

xn and

y2

,

. . . ,

yj−1

,

y1

,

yj+1

,

. . . ,

yn are proper interval orderings for G

−

x1. In both cases, we conclude the claim by induction. Note that G

−

x1 is indeed a connected proper intervalgraph.

Proofof2)

Ifx1

=

y1 thentheclaimtriviallyholds,ifx1xn

∈

E

(

G

)

thenG isacompletegraph,andtheclaimholds.So,astheremaining case, assume that x1

=

y1 and x1xn

∈

/

E

(

G

)

. Assume x1

≺

τ xn. Let p besuch that xp

=

y1. Recallthat

(

xp

,

. . . ,

xn

)

isan xp

,

xn-pathofG.Foracontradiction,supposethatx1y1

∈

/

E

(

G

)

.Then,

(

xp

,

. . . ,

xn

)

doesnotcontainanyvertexfromNG[x1

]

, since NG

[

x1

]

⊆ {

x1

,

. . . ,

xp−1

}

. We consider

τ

. Let P

= (

u0

,

. . . ,

ur

)

be an arbitrary y1

,

xn-pathof G.Since y1

≺

τ x1

≺

τ xn and u0

=

y1 and ur

=

xn, thereis an index i with 0

≤

i

<

r such that ui

τ x1

≺

τ ui+1.Thus, x1 is avertex on P or P containsaneighbourofx1,whichfollowsfromthedeﬁnitionofproperintervalorderings,sothat P containsavertexfrom NG

[

x1

]

.Bythe choice of P as an arbitrary y1

,

xn-path of G,it followsthat every such pathof G containsa vertexfrom NG

[

x1

]

,contradictingtheinitialassumption.Thus,x1y1

∈

E

(

G

)

.Theresultsfromthebeginningoftheproofthenshowthat NG

[

x1

]

⊆

NG

[

y1

]

andNG

[

y1

]

⊆

NG

[

x1

]

,i.e.,NG[x1

]

=

NG[y1

]

.

Theremainingproofforthesituationofxn

≺

τ x1 followsfromapplyingtheﬁrstsituationto

σ

andthereverseof

τ

.

2

Lemma 4.1readilyimpliesthefollowing.

Proposition 4.2. Let G be a connected proper interval graph with proper interval orderings

σ

and

τ

. Then,

σ

is strongly neighbourhood-equivalentto

τ

ortothereverseof

τ

.

We now prove anotherimplication ofLemma 4.1,that will play acrucial role inthe proof ofthe main resultofthis section.

Lemma4.3.LetG andH beproperintervalgraphswhereH isconnected.Let

σ

and

τ

beproperintervalorderingsforrespectivelyG andH ,andlet

τ

R_be_the_reverse_of

_τ

_._Then,_{H is}_isomorphic_to_an_induced_subgraph_of_{G if}_and_only_if_there_is

_ω

_{∈ {}

_τ

_,

_τ

R

_}

_such_that H is

(

σ

,

ω

)

-isomorphictoaninducedsubgraphof G.

Proof. If H isa graphon asingle vertex,theclaim triviallyholds.We thereforeassumethat H hasatleasttwovertices. Clearly, if H is

(

σ

,

τ

)

- or

(

σ

,

τ

R

₎

_-isomorphic_to_an _induced _subgraph_of _{G then} _{H is} _isomorphic_to_an _induced _subgraph of G.

Fortheconverse,assumethatG hasaninducedsubgraph Gsuchthat H isisomorphictoG;let

ϕ

beanisomorphism from H toG.ObservethatGisconnected.Let

σ

betherestrictionof

σ

totheverticesofG.Observethat

σ

isaproper interval orderingforG.Assumethat

σ

=

x₁

,

. . . ,

xrand

τ

=

y1

,

. . . ,

yr.Recall that

τ

R

=

yr

,

. . . ,

y1

andx1

≺

σ

· · · ≺

σ x_r.Let

ϕ

:

V

(

H

)

→

V

(

G

)

bethemappingwhere yi ismappedto xi andlet

ϕ

:

V

(

H

)

→

V

(

G

)

bethemappingwhere yi ismapped tox_r₊₁₋_i.Weshowthat oneofthetwomappingsisan isomorphismfrom H to G,whichprovestheclaimed result.

First, assumethat

ϕ

(

y1

)

≺

σ

ϕ

(

yr

)

.We showthat

ϕ

isan isomorphismfromH to G,byshowingforevery vertex y of H that NG

[

ϕ

(

y

)

]

=

NG

[

ϕ

(

y

)

]

. Let

σ

∗

=

def

ϕ

(

y1

),

. . . ,

ϕ

(

yr

)

. Observe that

σ

∗ is a proper interval ordering for G, and

ϕ

(

y1

)

≺

σ∗

ϕ

(

yr

)

. Recallthat

σ

is alsoa proper interval ordering for G, andthat

ϕ

(

y1

)

≺

σ

ϕ

(

yr

)

. Hence, we can apply Lemma 4.1to ﬁndthat

σ

∗ and

σ

are stronglyneighbourhood-equivalent.Since x_i

=

ϕ

(

yi

)

forevery i

∈ {

1

,

. . . ,

r

}

, we obtainthedesiredresult.If

ϕ

(

yr

)

≺

σ

ϕ

(

y1

)

then ananalogousproofshowsthat

ϕ

isan isomorphismfromH to G, whereLemma 4.1isappliedto

ϕ

(

yr

),

. . . ,

ϕ

(

y1

)

and

σ

.

2

Wearenowreadytogivethemainresultofthissection.

Theorem4.4.Givenaproperintervalgraph G andaconnectedgraph H ,itcanbedecidedinpolynomialtimewhetherG hasan inducedsubgraphthatisisomorphictoH .

Proof. We describesuch analgorithm.First, assumethat G and H are connectedproperinterval graphs.Let

σ

and

τ

be proper interval orderings forrespectively G and H . Let fG and fH be 1-colourings forrespectively G and H . Let

τ

R be the reverseof

τ

.We runAlgorithm OIS on

(

G

,

fG

,

σ

;

H

,

fH

,

τ

)

andon

(

G

,

fG

,

σ

;

H

,

fH

,

τ

R

)

andaccept ifandonlyifOIS acceptson(atleast)one ofthetwoinputs.IfH isnotaproperintervalgraph,reject,andifG isnotconnected,applythe aboveproceduretoeveryconnectedcomponentofG.

Letusprovethe correctnessofthealgorithm.As properinterval graphsare hereditary,wecan correctlyrejectif H is not a proper interval graph.ByLemma 4.3, H is isomorphicto an induced subgraph of G if andonlyif H is

(

σ

,

τ

)

- or

(

σ

,

τ

R

₎

_-isomorphic_to_an _induced_subgraph_of_G._Proper_interval_orderings _together_with_any_colouring_satisfy_the_left_and right umbrellaconditions,inparticular,forthe 1-colourings usedinthe algorithm.It followsfromCorollary 3.4that our algorithmwillacceptifandonlyifH isisomorphictoaninducedsubgraphofG.