THE DOMATIC NUMBER PROBLEM IN INTERVAL-GRAPHS

THE DOMATIC NUMBER PROBLEM IN INTERVAL GRAPHS*

TUNG-LIN LUg’, PEI-HSIN HO’, AND GERARD J.

_CHANGer

Abstract. AsetofverticesDis adominatingsetofagraph G (V,E)if every vertex inV Disadjacent

toa vertexinD.The domatic numberd(G)ofagraph G (V,E)is the maximum numberk such thatVcan

be partitioned intok disjoint dominatingsetsD, Dk.The main purpose of this paper is togivelinear algorithmsfor thedomaticnumber problemin intervalgraphs.This paperalsoproves thatd(G) 6(G) +

forany intervalgraphG,where6(G)isthe minimumdegree ofa vertex inG.

Keywords, dominating set,domaticnumber,intervalgraph, degree,linear algorithm, NP-complete

AMS(MOS)subject classifications. 05C70, 68R10

1. Introduction. A set of verticesD is adominating setofagraph G (V, E)if

everyvertex in V-Disadjacentto a vertex inD.The domaticnumber

d(G)

ofagraph

G (V, E) isthe maximum number ksuch that Vcan bepartitioned into k disjoint dominatingsetsD1,

Dk.

Thedomination set problemanditsvariationshavebeen extensivelystudied; however, thedomaticnumber problemismuch lesswell known.

Lowerboundsand upperbounds for thedomaticnumber were studied in 4]- 6], [9]-[1 1], and

[13].

Inparticular, [6] showed thatd(G) _-< _t3(G)

₊

forany graph G,

where 6(G) isthe minimum degree ofa vertex in G. Gis domatically

_full

if

d(G)

6(G)

+

1. Cockayne andHedetniemi [6] determined

d(G)

forsome special classes of graphs; consequently, Kn, Kn, C3,, trees and maximal outerplanargraphs are

domati-callyfull.

The domatic numberproblem is NP-completefor general graphs [7]and circular-arc graphs [2]. The problem is solved in O(n2_log

n) time for proper circular-arc graphs 2], O(n

25)

timefor interval graphs and O(n logn)timeforproperinterval graphs

].

The main purposeofthis paper is togivelinearalgorithms forthe domatic number

problemin intervalgraphs.Asaby-productwealso provethatintervalgraphsare

domat-ically full.

An intervalfamilyis a setof intervals onthe real line.Aninterval family isproper if nointervalisproperly containedwithinanotherinterval.Agraphis a(proper) interval graphifthereis a one to one correspondence between theverticesofthegraphand the

intervalsofa(proper)intervalfamily suchthat two vertices arejoined byanedgeif and

onlyif theircorrespondingintervalsoverlap.

Interval graphs have been extensively studied and used as models for many real

world problems. Inparticular, they haveapplications in archaeology, genetics, ecology, psychology, trafficcontrol,computerscheduling, storageinformationretrieval,and

elec-troniccircuit design(see 8], 12]).

Booth and Lueker 3 gavealinear algorithm for deciding whether agiven graph

is an intervalgraphandconstructing,in theaffirmative case, therequired intervalfamily.

Inthispaper webegin with the assumptionthatGis known tobean intervalgraph, and acorresponding interval familyisgiven.

Receivedby theeditorsJune30,1989;acceptedforpublication(inrevisedform)March6, 1990.This

researchwassupported bythe NationalScienceCouncilofthe RepublicofChina under grant

NSC-77-0208-M009-21.

"t"

Department ofAppliedMathematics,National ChiaoTungUniversity, Hsinchu30050, Taiwan,Republic

ofChina.

531

2. Notation and assumptions. Suppose I

{1,..-, n}

is the interval family for an interval graph G produced by the linear algorithm in 3], where interval is equalto [a;,

b;]

for 1, n. _aiis the

left

endpoint ofinterval and

bi

the right endpoint. Withoutloss of generalitywemay assume that

{

a, an, b,

bn)

{1,2, "-,2n}.

Forthesakeof simplicitywe willdenoteanintervalgraphGasG(I)and dealwith

intervals insteadofvertices.Inthisway, theclosedneighborhoodN[i] ofaninterval is

thesetof allintervalsthat overlapwithintervali.AdominatingsetforG(I)corresponds

to a subset Sofintervals in Isuch that every interval in Ioverlaps with at least one

interval in S.

Foreach interval next(i) istheinterval jsuch that

_ba.

is assmall as possible but

satisfying

b

<

a;

next(i)isnullif no such jexists.

3. The algorithms. Inthis section we willgivetwo efficient algorithms forthe

do-maticnumberprobleminintervalgraphsG(I). Fortechnicalreasons, wefirst augment Iwith two"dummy" intervals0 andn

+

such that a0 -1,

b0

0,

an+

2n

+

and

_bn

+ 2n

+

2.Wethenconstruct anacyclic directed graphHasfollows. The nodes

ofHcorrespond totheintervals in

I’

ItO

{

0, n

+

}.

There is a directed arc(i, j)in Hif and only if j N[next(i)]. Note that if(i, j) is an arc in

H,

then

bi

<

_ba..

This

guarantees thatHis acyclic. Since

N[

n

+

n

+

},

(i, n

+

is an arc inHif and

onlyif next(i) n

+

1.

LEMMA 3.1. Anydirected path

from

node 0 tonode n

+

in Hcorresponds to a

dominatingset

for

G(I).

Proof.

Suppose io,il,

ir)

is a directedpathfrom node 0tonode n

+

inH. Bythedefinitionofanarcin

H,

_bi0

<

bi,

< <

_bir.

Foranyintervalj

I,

choosean index ssuchthat

bi,_

<

a;

<

bi,.

Supposeintervals j and i,donotoverlap. Then

bi,_,

<

a

<

_b

< ai,<

bi,.

Let k next(i,_1). Bythe definitionoffunctionnext,

b

=<

b

and so

intervalskand i,donot overlap.That implies i, N[next i,_ ],oncontradicting that (i,_1, i,)is an arc inH. Thereforej eN[i,] and so il, it_

}

is adominatingset forG( I). 73

Adominating set of

G(I)

does not necessary correspond to a directed path from node 0 to node n

+

in H. So we cannot conclude immediately, as in [1 ], that the domaticnumberofG(I)isequalto themaximumnumberof disjoint pathsfromnode

0to noden

+

inH. InfactourdefinitionofdirectedgraphHis differentfromthat in

[1

]. The wayouralgorithmswork isby meansofthefollowing dualityrelation.

LEMMA 3.2 [6] (weakduality inequality),

d(G)

_-<_6(G)

₊

_for

_anygraph G.

The mainidea ofouralgorithmsis tofind6

+

disjoint dominating setsin G(I), orequivalently 6

₊

disjoint paths from node 0 tonode n

+

inH. We will present two algorithms for this purpose. The first algorithm finds the dominating sets one by

one.The second algorithm findsalldominatingsetssimultaneously.Section 4 implements

these algorithms andshows that their running times arelinear.

ALGORITHM D

initially allintervals areunlabeled;k

-

0; loop

--

0;

while(next(i) 4 n

+

d__0_o

j

--

next(i);

i_fN[j] has no unlabeledintervalsthen

STOP;

choose anunlabeled interval h N[j] withlargest leftendpoint;

labelh byk

₊

1;

i--h;

endwhile; k---k+ 1; forever.

Suppose k*isthe finalkwhenAlgorithmD stops.Let

_Di

be thesetof allintervals

labeled by i.By Lemma 3.1, wehavek* disjointdominatingsetsD,

Dk.

LEMMA 3.3. Thereexistsan intervalj such that IN[j]I k*.

Proof.

WhenAlgorithmD stopsthereexists some interval j’such that allintervals inN[j’] arelabeled by integersbetween and k*. Letjbe suchaninterval with largest leftendpoint.

SupposeN[j]containstwodistinctintervals pandqof thesame label k.Assume intervals in

_Dk

arelabeledintheorder. .p p,P2, Pm q, Bythedefinition

of function next,

(3.1)

bp=bp<anext(pl) <bp2<anext(p2)<’" <bpm_l<anext(pm_)<bpm--bq.

Alsoaj<

bp

since p N[j].Letr next(pm_1).Thenaj<

ar.

Bythe choice ofj,N[r]

has an interval swhich is unlabeledor islabeled by k*

+

1. Suppose

_as

> aq. Sinces,

q N[ r], by Algorithm D1, interval s would be labeled by k before interval qbeing

labeledby k. So

as

< aq. Alsoaq <

_b

since q N[j]. Then

as

<

_b.

Ontheotherhand,

aj<bp<ar<bs.

The firstinequality follows fromthat p N[j],the second is partof 3.1),andthe third

from s 6 N[ r]. Both

as

<

_b

and aj <

bs

imply that s 6 N[j], a contradiction to the assumptionthatall intervals in N[j] are labeledby integersbetween andk* but that

s isnot. Henceall intervals in N[j] havedistinctlabels, i.e., [N[j]] k*. Vq

ByLemmas 3.2 and 3.3we have

/5+l=min IN[i][=< _{IN[j][ =k*<=d(G(I))<=6+ 1,} iI

andso in fact theabove inequalities are equalities.

THEOREM 3.4 (strong duality theorem),

d(G)=

6(G)

+

for

any interval

graph G.

THEOREM 3.5. Algorithm D1 works

for

solving the domatic number problem in

intervalgraphs.

The second algorithm follows fromthefactthat the outdegree of eachnode inH

such that (i, n

+

is not an arc inHis IN[next(i)][ >- 6

+

1. We will describe our

algorithmin termsofHhereand implement it in termsofintervalsinthenextsection.

ALGORITHM D2

initially all nodesinHareunlabeled;

find atopologicalsortio 0,ii, i, i,

i+1

n

+

forthe nodesofH

(i.e., (ip, iq)is an arc inHimpliesp <q);

label thefirst rintervals (according tothe topologicalsort) inN[next(0) by 1, rrespectively;

forr

--

to n do

i_f ir, n

+

is not an arc inHand

ir

is labeledby kthen

* choose anunlabeled nodej with ir,j)is an arc inHand labeljbyk; end for.

Since io 0, l, i2, i,

i

/ n

+

is atopological sortfor the nodes of

H,

the factthateach node ofHsuch that i,n

+

is not an arc inHhas outdegreeatleast

6

+

impliesthateach subgraph

_Hr

ofHinducedby

{

it,

ir+

1,

i

+

}

has thesame

property. Soin Step(*)of Algorithm D2,we canalwaysfind such anodej.Foreachk between and 6

+

1,

{

0, n

+

togetherwith all nodeslabeled byk form a directed

path from node0tonode n

+

1. Thealgorithmdoesproduce6

₊

disjointpathsfrom node 0 to node n

+

1. So we have another way to verify Theorem 3.4 and solve the domaticnumberproblem.

THEOREM 3.6. Algorithm D2 works

_for

solving the domatic number problem in

intervalgraphs.

4. Implementation. Thissection givestwo implementationsfor AlgorithmD and one implementationforAlgorithm D2. The implementations showthatthealgorithms

arelinear.

Weassume

_{

al, an, b,

bn

{

1, 2, 2n

}.

We alsoneed the

infor-mationthat eachp e 1,2n] isequalto

_a

or

_b

forsomeinterval eI.This canbe done

byasimple do loop. Wecan use one array toindicatethat p is aleftor afight endpoint

andanother array to indicate that p is the endpointofaninterval I.

To implement our algorithms, we first must produce function next. This can be done in O(n)time asfollows. The algorithm scans the endpointsfrom 2n backward to -1.In the algorithm, s is the interval with smallestfightendpoint among the intervals

whose leftendpoints havebeen scanned.

//*Calculate

function next*

/ /

s--n+

1;

four

p

--

2n to step do

case 1" p aiforsomeinterval e

I’

i_f

bi

<

b

then s

-

i;

case2: p

_bi

forsomeinterval e

I’

next(i)

--

s; endfor.

Havingcalculated function next, we nowpresent the first implementation of

Al-gorithmD1.The maindifficultyinAlgorithmD ishow to "choose anunlabeledinterval heN[i] withlargest leftendpoint."Weuse astackto storecandidates ofsuch intervals

such thatan intervalwithlarger leftendpoint isclosertothetopofthe stack. ] ]*First implementation of AlgorithmDI*] ]

label(i)

-

0forall intervals e

I;

k

-

0;

loop

stackS

-

qS;h

--

0; forp

--

to2n d__o

j

-

next(h);

case 1" p aiforsome interval I.

i_f label (i) 0 then push intoS; case2: p

_bi

forsomeinterval I.

i_f jthen

while(S4: 4) do pop hfromS;

i_f

_bh

> ajthen

[label(h)

--

k

+

1;

i_fnext

(h)

n

+

then goto * * _{els__e goto} *

endwhile;

STOPthe algorithm; end if;

* _{end, for;}

(**)k-k+

1;

forever.

Asintheproof afterLemma3.3, thereare 6

₊

2iterations inthe loop. Each iteration

needs

O(n)

steps. So thisis an O(rn

+

n)

0(I

El +

[VI)

time implementation of Algorithm D1.

Asasecondimplementation, weonlyhave toproduce"sorted"closed neighborhoods

N[i] for all intervals I. Oncewehave sorted closed neighborhoods,AlgorithmD is

easilyimplemented. Wenowgivean

O(1

El +

VI)

algorithm for producingtheclosed

neighborhoodof allintervals inIasfollows.

]/*Each

closed neighborhoodN[ i/issorted according to left endpoints*

//

doublylinkedlistL

--

4);

forp

--

to2nd__0_o

case 1" p aiforsomeinterval I

N[

i]

--

L;

L

--

L

+

i;//*remembertheaddressof inL*/ /

for eachj6Ld__0_oN[j]

-

N[j]

+

i; case2: p

_bi

forsomeinterval 6I

delete from

L;

end for.

Finally, we shall implement Algorithm D2 in terms ofintervals rather than the directedgraph H. Note thatif (i, j) is an arc in

H,

then

bi

< bj. A simple topological sortof the nodes ofHis to sort intervals inI’according to their right endpoints.

//*Implementation of algorithmD2*

/ /

label(i)

-

0for all intervals

I;

p

--

1;

fork- ltor+ ldo

i_fp _aifOrsomeinterval which intersects interval next(0)then label (i)

-

k;

p--p+

1;

endfor;

forp

--

to2n d__o

i_fp

_bi

forsome interval Ithen

i_flabel(i) 4:0 andnext(i) 4: n

+

then [getj N next(i)] with label(j) 0;label(j)

-

label(i)] end for.

Inthe above implementation,each closed neighborhoodis considered as anunsorted singlelinked list. Toget someintervalj inN[next(i)]wesimplygetthe first element of the list and delete it from the list. Ifits label is nonzero, we continue to get the first

element from the remaininglist until an interval withlabelzero isfound.

5. Properintervalgraphs. InthecasewhereG(I)is a properinterval graph,assume

a

<a2< <

an.

Itis easy to see thateach closedneighborhoodN[i] is a consecutive

set, i.e.,N[i] j, j

+

1, j

+

k

}

forsome j andk. Sinceeach

[N[

i]l

>--

6

₊

1,we can construct 6

+

disjoint dominating sets

Dk

{

I: kmod 6

+

},

k 1,

6

+

1. This is a much simplermethodthan that in[1 ].

6. Conclusion. The main results ofthis paper give two linear algorithms for the

domaticnumberproblemoninterval graphs.Asaby-productwealsoshow that interval

graphsaredomaticallyfull. Amuch simplermethod fortheproblem inproperinterval graphsis also discussed.Wesuspectthatsimilar results canbeobtainedfor the problem

instrongly chordal graphs oreven for chordal graphs.

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