Grippo, Luciano Norberto

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Tesis Doctoral

Caracterizaciones estructurales de

grafos de intersección

Grippo, Luciano Norberto

2011

Este documento forma parte de la colección de tesis doctorales y de maestría de la Biblioteca

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acompañada por la cita bibliográfica con reconocimiento de la fuente.

This document is part of the doctoral theses collection of the Central Library Dr. Luis Federico

Leloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the corresponding

citation acknowledging the source.

Cita tipo APA:

Grippo, Luciano Norberto. (2011). Caracterizaciones estructurales de grafos de intersección.

Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires.

Cita tipo Chicago:

Grippo, Luciano Norberto. "Caracterizaciones estructurales de grafos de intersección". Facultad

de Ciencias Exactas y Naturales. Universidad de Buenos Aires. 2011.

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UNIVERSIDAD DE BUENOS AIRES

Facultad de Ciencias Exactas y Naturales

Departamento de Matem´atica

Caracterizaciones Estructurales de Grafos de Intersecci´on

Tesis presentada para optar al t´ıtulo de Doctor de la Universidad de Buenos Aires en el ´area

Ciencias Matem´aticas

Luciano Norberto Grippo

Director de tesis: Dr. Guillermo Alfredo Dur´an

Director Asistente: Dra. Flavia Bonomo

Consejero de estudios: Dra. Claudia Lederman

Lugar de trabajo: Departamento de Computaci´on, FCEyN, UBA.

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Enestatesisestudiamosaraterizaionesestruturalesparagrafos

aro-irulares,grafosírulo,grafosprobe deintervalos,grafosprobe de

interva-losunitarios,grafosprobe debloquesygrafosprobe o-bipartitos. Ungrafo

esaroirular (írulo)sieselgrafodeinterseióndeunafamilia dearos

(uerdas) en una irunferenia. Dada una familia hereditaria de grafos G,

un grafo es probe G si sus vérties pueden partiionarse en dos onjuntos:

unonjunto devértiesprobe yunonjuntodevérties nonprobe,de forma

tal que el onjunto de vérties nonprobe es un onjunto independiente y

esposible obtener un grafoen la lase G agregando aristas entre ellos. Los

grafos probe G forman una superlase de la familia G. Por lo tanto, los

grafos probe de intervalos y los grafos probe de intervalos unitarios

gener-alizan la lase de losgrafos deintervalos ylos grafosde intervalos unitarios

respetivamente.

Caraterizamos parialmente a los grafos aro-irulares, grafos írulo,

grafosprobe deintervalosyprobe deintervalo unitariomediantesubgrafos

prohibidos dentrode iertas familiashereditarias de grafos. Finalmente, es

presentada una araterizaión de los grafosprobe o-bipartitos quelleva a

unalgoritmode reonoimiento detiempopolinomial para dihalase ylos

grafosprobe de bloques sonaraterizados mediante unalista de subgrafos

prohibidos.

Palabras lave: grafos aroirulares, grafos írulo, subgrafos induidos

prohibidos,grafosprobe debloques,grafosprobe o-bipartitos, grafosprobe

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In this Thesis we study strutural haraterizations for six lasses of

graphs, namely irular-ar graphs, irle graphs, probe interval graphs,

probe unit interval graphs, probe o-bipartite graphs, and probe blok

graphs. A irular-ar graph (irle graph) is the intersetion graph of a

familyofars (hords)on airle. LetG beahereditary lassof graphs. A

graphisprobeGifitsvertiesanbepartitionedintotwosets: asetofprobe

verties and aset ofnonprobeverties,sothattheset ofnonprobe verties

is a stableset and it is possible to obtain a graph belonging to thelass G

byaddingedges withbothendpointsinthesetofnonprobeverties. Probe

G graphsformasuperlass ofthelassG. Hene, probeinterval graphsand

probe unit interval graphs are extensions of the lasses of interval graphs

and unit interval graphs,respetively.

We partially haraterizeirular-ar graphs, irle graphs,probe

inter-val graphs and probe unit interval graphs by forbidden indued subgraphs

withinertainhereditaryfamilies ofgraphs. Finally,astrutural

harater-izationforprobeo-bipartite graphsthatleads to apolynomial-time

reog-nition algorithm and a ompleteharaterization of probeblok graphs by

alist of forbiddenindued subgraphsarepresented.

Keywords: irular-argraphs,irlegraphs,forbiddeninduedsubgraph,

probeblokgraphs,probeo-bipartite graphs,probeinterval graphs,probe

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AmisdiretoresFlaviayWilly,nosoloportodoelempujequeledieron

a este trabajo de tesis si no por permitirme partiipar de este grupo de

investigaión de Teoría de Grafos y Optimizaión donde me siento muy a

gusto.

Al Departamento de Computaión, por habermedado un lugar de

tra-bajo. AlDepartamento deMatemátia, dondemeformé. A Claudia

Leder-man, mi onsejerade estudios.

AlICI,dondeonoígenteonunainmensaalidadhumana. Enespeial

alDTA,alÁreadeMatemátiaeInformátia,aCristianyamisinigualables

ompañerosde oina Ezequiel yMariano.

A mis amigos de la vida Alejandro, Andrés, Javier y Mauro. A mi

hermanoaadémioyamigoMartín. AAndrésPortaySantiagoGuaraglia.

A Liliana y Raúl a quienes quiero muho. A mis queridos padres

Nor-bertoyNorma portodo suapoyo yamor.

A Laura,la mujerqueamoya quiénestámereidamentedediada esta

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Introduión

Los grafos aro-irulares son los grafos de interseión de una familia

S de aros en una irunferenia, al onjunto S se lo llama modelo

aro-irular. Los primerostrabajossobre estalase de grafosfueron publiados

porHadwigeryotrosen1964[HDK64 ℄yporKlee[Kle69℄en1969

respetiva-mente. Sinembargo, el primeroen trabajar en el problema de araterizar

por subgrafos prohibidos esta familia de grafos fue A. Tuker en su tesis

dotoral en 1969 [Tu60℄. Fue él mismo quien introdujo y onsiguió

ar-aterizar por subgrafos prohibidos dos sublases de grafos aro-irulares:

grafos aro-irulares unitariosy grafosaro-irulares propios. Laprimera

sublase onsiste en aquellos grafos aro-irulares que poseen un modelo

aro-irular on todos sus aros de la misma longitud y la segunda

sub-lase son los grafos aro-irulares on un modelo donde ningún aro está

ontenido en otro.

Caraterizar la lase ompleta de grafos aro-irulares por subgrafos

prohibidos esun problema abierto desde haemuho tiempo. Sinembargo

varios autores han presentado algunos avanes en esta direión. Trotter y

Mooredieronunaaraterizaión porsubgrafosprohibidosinduidosdentro

de lalase de grafoso-bipartitos [TM76 ℄. J.Bang-Jensen yP.Hell

presen-taron un teorema estrutural paragrafos aro-irulares propios dentro de

la lase de grafos ordales [BH94 ℄, del ual se desprende la araterizaión

por subgrafos induidos prohibidos para los grafos aro-irulares propios

restringidos ala lase de los grafos ordales.

Los grafosaro-irulares sonuna generalizaión de la familia de grafos

de interseión de intervalos en la reta real,llamados grafos de intervalos.

Los grafos de intervalos fueron araterizados por Boland y Lekkerkerker

en 1962 [LB62℄. La lista ompleta de subgrafos induidos prohibidos que

araterizalosgrafosdeintervalosfuehalladaexitosamenteviauna

arater-izaiónpormediodetriplas asteroidalespresentada porlosmismosautores.

Todo onjunto deintervalos en lareta realsatisfae la propiedadde Helly;

es deir, ualquier onjunto de intervalos mutuamente interseantes en la

reta real tiene un punto en omún. Por lo tanto una sublase de grafos

aro-irulares que generaliza a los grafos de intervalos de forma natural

sonlos grafos aro-irulares Helly; es deir, aquellos grafos aro-irulares

que tienen un modelo que satisfae la propiedad de Helly. Lin y

Szwar-ter presentaron una araterizaión para esta lase mediante estruturas

prohibidas dentro de la lase de los grafos aro-irulares [LS06a℄. Diha

araterizaión lleva a un algoritmo de reonoimiento lineal para la lase

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la lase de los grafos aro-irulares propios Helly [LSS07 ℄, aquellos grafos

quetienenun modeloaro-irularqueessimultáneamentepropioyHelly.

P.Hellprobóquelafamiliadelosbigrafosdeintervalossonexatamente

aquellosgrafos aro-irulares on númerode ubrimiento porlique dos y

poseen un modelo aro-irular sin dos aros que ubran la

irunferen-ia ompleta. Los grafos aro-irulares que satisfaen diha ondiión son

onoidos enlaliteraturaomo grafosaro-irulares normales. Esta

termi-nologíafueintroduidaen[LS06b ℄. Generalizandolosgrafosaro-irulares,

L.Alón yotros introdujeron la lase de losgrafos bule.

A pesar de que muhos investigadores han tratado de enontrar la lista

desubgrafosprohibidosquearaterielalasedelosgrafosaro-irulares,

elproblema aún permanee abierto. En esta tesis presentamos algunos

pa-sos en esta direión, aportando araterizaiones de grafos aro-irulares

por subgrafos induidos prohibidos minimales uando el grafo pertenee a

algunadelassiguienteslases: grafossinP

4

,grafossin paw,grafosordales

sin law y grafos sin diamante. Además, omo los grafos aro-irulares

que perteneen a estas lases tienen un modelo aro-irular normal, estos

resultadosimplian que los subgrafosinduidos prohibidos parala lase de

los grafos aro-irulares normales neesariamente ontienen un diamante

induido,un P

4

induido,un paw induido,yo bien unlaw oun agujero

omosubgrafoinduido. Tambiénintroduimosyaraterizamoslalasede

los grafos semiirulares, grafos aro-irulares que tienen un modelo

aro-irular dondetodos sus aros son semiirunferenias. Cabe destaar que

todasestaslasesfueronestudiadasalolargodelaminohaialapruebadel

TeoremaFuertedelosGrafosPerfetos[Con89 ,Ola88,PR76,Sei74 ,Tu87 ℄.

Un grafo sedie írulo si esel grafode interseión de un onjunto de

uerdasenuna irunferenia, atalonjunto selollamamodelo deírulo.

LosgrafosírulofueronintroduidosporEveneItaien[EI71 ℄pararesolver

unproblemadeordenamiento onelmínimonúmerodepilasenparalelosin

larestriióndeargarantesquela desargaseaompletada. Ellostambién

probaron que este problema se puede traduir en el problema de hallar el

número romátio de un grafo írulo. Desafortunadamente este problema

resultaser NP-ompleto [GJMP80℄.

Naji araterizó los grafos írulo en términos de la soluión de un

sis-temalinealdeeuaiones quellevaaunalgoritmode reonoimiento O

(

n

7

)

para esta lase [Naj85℄. El omplemento de un grafo G on respeto a un

vértie u 2 V

(

G

)

es el grafo Gu que se obtiene a partir de G

reem-plazando el subgrafo induido G

[

N

G

(

u

)]

por su omplemento. Este tipo

de operaión se denomina omplementaión loal. Se die que dos grafos

G yH son loalmente equivalentes si y solo si G se obtiene a partir de H

mediante una suesión de omplementaiones loales. Bouhet probó que

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queungrafoesírulosiysolositodografoloalmenteequivalenteno

on-tiene tres determinados grafos omo subgrafo induido [Bou94℄. Geelen y

Oum[GO09 ℄dieronunanuevaaraterizaión degrafosíruloen términos

delaoperaióndepivoteo. ElresultadodepivotearungrafoGonrespeto

a una arista uv es la grafo Guv

=

Guvu (donde representa a la

omplementaión loal). Un grafo G

0

es equivalente por pivoteo a G si G

0

seobtienea partirde Gmedianteuna seuenia de operaiones de pivoteo.

Ellosprobaron,on laayudadeunaomputadora,queGesungrafoírulo

si y solo si ada grafoequivalente por pivoteo a Gno ontiene ninguno de

15 grafosdeterminados omo subgrafoinduido.

Un grafoírulo que posee un modelo talque todassus uerdas tienen

la mismalongitud se llama grafo írulo unitario. La lase de grafos

aro-irulares propios está propiamente ontenida en la lase de los grafos

ír-ulo. Más aún, la lase de los grafos aro-irulares unitarios oinide on

la lase de los grafosírulo unitario.

Deimos que Gtiene una desomposiión split si existendos grafos G

1 yG 2 on jV

(

G i

)

j3,i

=

1,2,talque G

=

G 1 G 2 on respeto aalgunos

vérties destaados(verCh. 3 ). Siestosuede llamamosaG

1

yG

2

fatores

de la desomposiión split. A aquellos grafos que no poseen una

desom-posiión split selos llama primos. El onepto de desomposiión split es

debido a Cunningham [Cun82 ℄. Los grafos írulo resultaron ser errados

pordesomposiión split [Bou87℄y en 1994 Spinradpresentó un algoritmo

de tiempo uadrátio que seaproveha de esta peuliaridad.

Los grafos írulo son una superlase de los grafos de permutaión. De

heho, losgrafos de permutaión puedenser denidosomo aquellos grafos

írulo tales que una uerda que interseque todas las uerdas del modelo

puede ser agregada. Por otro lado los grafos de permutaión son aquellos

grafos de omparabilidad uyo grafo omplemento es también de

ompa-rabilidad. Como los grafos de omparabilidad han sido araterizados por

subgrafos prohibidos induidos en [Gal67℄, tal araterizaión implia una

araterizaiónporsubgrafosinduidosprohibidosparalalasedelosgrafos

de permutaión.

Los grafos de írulo Helly sonaquellos grafos de írulo que tienenun

modelo uyas uerdas satisfaen la propiedad de Helly; es deir, todo

on-junto de uerdas que se intersean dos a dos tienen un punto en omún.

Esta familia de grafos fue introduida por Durán en [Dur00℄. Él también

onjeturóqueungrafoíruloesíruloHellysiysolosinoontiene un

dia-manteomosubgrafo induido. Reientemente, éstaonjeturafueprobada.

Sin embargo, los grafos írulo Helly aún no han sido araterizados por

subgrafosprohibidos.

En esta tesis presentamos algunas araterizaiones pariales por

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grafos domino lineales usando la desomposiión split. Conseuentemente,

araterizamos los grafos írulo Helly dentro de la lase de los grafos sin

law. Caraterizaiones por subgrafos induidos prohibidos dentro de dos

superlasesde ografos,tree-ographs yP

4

tidy,sonpresentadasomouna

apliaióndelaaraterizaión deGallaiparagrafosdeomparabilidad.

Fi-nalmente introduimosy araterizamos la lase de los grafos írulo Helly

unitarios, aquellos grafos írulo que tienen un modelo que es

simultánea-mente Helly yunitario.

Sea G una lase hereditaria de grafos. Un grafo se die probe G si sus

vérties puedenserpartiionados endosonjuntos: un onjunto de vérties

probe y un onjunto de vérties nonprobe, de forma tal que el onjunto

de vérties nonprobe es un onjunto independiente y se puede obtener un

grafoperteneiente ala lase Gagregando aristas onambosextremosen el

onjunto de vérties nonprobe.

En 1994 Zhag introdujo los grafos probe de intervalos omo una

he-rramienta de investigaión en el maro del proyeto del genoma humano,

[ZSF

+

94 ℄. Desde entones, los grafos probe G han sido estudiados para

diferentesfamiliashereditarias degrafosG. Shengaraterizóporsubgrafos

induidos prohibidos aquellos árboles que son probe de intervalo [She99 ℄.

Brownyotrospresentaronunaaraterizaión porsubgrafosinduidos

pro-hibidosdentrodelalasedelosárboles[BLS09 ℄. PrºuljyCorneilestudiaron

lossubgrafosprohibidosparagrafosprobe deintervalosdentrodelalasede

los2tree [PC05 ℄. Browny Lundgrenprobaronque los grafosprobede

in-tervalos bipartitossonequivalentesaelomplemento deuna lasedegrafos

aro-irulares on númerode ubrimiento lique dos[BL06 ℄. En [BBd09 ℄,

Bayeryotros araterizarondossublasesde losgrafosprobe de intervalos,

losgrafos probe threshold ylos grafosprobe trivialmente perfetos,en

tér-minosde iertas fórmulas2SAT. En elmismoartíulo ellos presentan una

araterizaión porsubgrafosprohibidos para los grafosprobe threshold.

Las lases de losgrafos probe G,on G diferentes delos grafos de

inter-valos yde intervalos unitarios, han sido estudiadas para importanteslases

de grafos omo por ejemplo: grafos ordales [GL04, CGLS10 ℄, grafos de

permutaión [CCK

+

09 ℄ygrafos split [LdR07℄entre otros.

Con el objetivo de estudiar el omportamiento del operador join para

losgrafos probe de intervalos e intervalo unitario introduimosel onepto

de lases hereditarias de grafos on un ompañero. También presentamos

la lista ompleta de todos los subgrafos induidosprohibidos uyo

omple-mento es disonexopara la lase de los grafos probe de intervalos. A pesar

deno onseguirhaer lomismoparala lase delos grafosprobe de

interva-losunitarios, presentamosla lista desubgrafos induidosprohibidos dentro

de la lasesde tree-ographs yP

4

tidy. Damos unaaraterizaión en

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que son probe de intervalos, esta araterizaión implia que los grafos

o-bipartitos probe de intervalos ylos grafos o-bipartitos probe de intervalos

unitariossonla mismalase. Losgrafos probe deintervalos yde intervalos

unitariossonaraterizadosporsubgrafosinduidosprohibidosdentrodela

lase de los tree-ographs, generalizando las araterizaiones presentadas

en [She99 ℄ y [BLS09 ℄, respetivamente. Finalmente, araterizamos por

subgrafosprohibidosinduidoslosgrafosprobe deintervalosydeintervalos

unitarios dentro de la lase de los grafos P

4

tidy. También estudiamos las

lasede grafosprobe de grafoso-bipartitos ygrafos debloques. Losgrafos

probe de bloques sonuna sublase de los grafos probe ordales estudiados

en [GL04, CGLS10℄. Los grafos probe ordales no han sido aún

arater-izados por subgrafos prohibidos induidos. En esta tesis presentamos una

araterizaión para grafos probe de grafos de bloques por subgrafos

pro-hibidos induidos yprobamos que la lase de los grafos probe de grafos de

bloqueseslainterseiónentrelaslasesdegrafosordalesyprobe degrafos

sindiamante. Tambiénpresentamosuna araterizaión estruturalparala

lase de los grafos probe o-bipartitos que lleva a un simple algoritmo de

reonoimiento de tiempopolinomial paraesta lase.

Esta tesis estáorganizada omo sigue. En elCapítulo 1 damosalgunas

deniionesyunbreveresumensobrelaslasesdegrafosestudiadas enesta

tesis. El Capítulo 2 está dediado a los grafos aro-irulares. En el

Capí-tulo 3 presentamos araterizaiones para grafos írulo. Los grafos probe

de intervalos ylos grafos probe de intervalos unitariossonestudiados en el

Capítulo 4 . En el Capítulo 5 presentamos araterizaiones estruturales

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Introdution xvii

1 Preliminaries 1

1.1 Denitions and notation . . . 1

1.2 Overview on somelasses ofgraphs . . . 3

1.2.1 Domino graphs . . . 3

1.2.2 Intersetion graphs . . . 4

1.2.3 Cographs and its superlasses . . . 8

2 Cirular-ar graphs 11 2.1 Introdution. . . 11 2.2 Preliminaries . . . 12 2.2.1 Denitions . . . 12 2.2.2 Previous results. . . 12 2.3 Partial haraterizations . . . 15 2.3.1 Cographs . . . 15 2.3.2 Paw-free graphs. . . 18

2.3.3 Claw-free hordal graphs. . . 20

2.3.4 Diamond-free graphs . . . 23

2.4 Summaryand furtherresults . . . 26

3 Cirle graphs 29 3.1 Introdution. . . 29

3.2 Linear domino graphs . . . 32

3.3 Superlasses ofographs . . . 37

3.3.1 P 4 -tidy graphs . . . 37

3.3.2 Treeographs . . . 38

3.4 Unit Helly irle graphs . . . 38

4 Probe interval graphs 41 4.1 Preliminaries . . . 41

4.2 Co-bipartite graphs . . . 43

4.3 Probe interval graphs . . . 46

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4.5 Partial haraterizationof probe f3K 1 ,C 4 ,C 5 g-free graphs . . 53

4.6 Partial haraterizations ofprobe unitinterval graphs . . . . 57

5 Probe o-bipartite and probe blok graphs 59 5.1 Probe o-bipartite graphs . . . 59

5.2 Probe blokgraphs . . . 60

5.2.1 Probediamond-free graphs . . . 60

5.2.2 Probeblok graphs. . . 68

6 Conlusions and future work 71

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Cirular-argraphs aretheintersetion graphsofasetS ofars onairle,

suh a set S is alled a irular-ar model. The rst works about this

lass of graphs were published by Hadwiger et al. in 1964 [HDK64 ℄ and

by Klee [Kle69℄ in 1969. Nevertheless, the rst researher who dealt with

theproblemof haraterizing byforbidden subgraphs this familyofgraphs

was A. Tuker in his Ph.D. Thesis in 1969 [Tu60 ℄. He introdued and

managed to haraterize by forbidden induedsubgraphs two sublasses of

irular ar-graphs: unitirular-ar graphsandproperirular-ar graphs.

Therst sublassonsists ofthose irular-ar graphs having airular-ar

model with allits ars havingthe same length and the seond one onsists

of those irular-ar graphs having a irular-ar model without any ar

ontained inanother.

Charaterizingbyforbiddeninduedsubgraphsthewholelassof

irular-ar graphs is a long standing open problem. Nevertheless, several authors

have presented someadvanes inthis way. Trotter and Moore gave a

har-aterizationbyforbidden induedsubgraphswithinthelassofo-bipartite

graphs [TM76 ℄. J.Bang-Jensen and P.Hell presenteda struturaltheorem

for proper irular-ar graphs within the lass of hordal graphs [BH94 ℄,

thatimpliestheharaterizationbyforbiddeninduedsubgraphsforproper

irular-ar graphs restrited to the lassofhordal graphs.

Cirular-ar graphsareageneralization ofthefamilyoftheintersetion

graphs of intervals in the real line, alled interval graphs. Interval graphs

wereharaterizedbyBoland andLekkerkerkerin1962 [LB62℄. The whole

list of forbidden indued subgraphs that haraterizes interval graphs was

suessfully foundvia aharaterizationbymeansof asteroidaltriples

pre-sented by the same authors. Any set of interval in the real line satises

the Helly property; i.e., any set of pairwise interseting intervals in the

real line have a ommon point. Consequently, a sublass of irular-ar

graphs that naturally generalizes interval graphs are theHelly irular-ar

graphs; i.e., those irular-ar graphs having an intersetion model of ars

suh thatany subsetofpairwise interseting ars hasaommon point. Lin

andSzwarterpresentedaharaterizationbyforbiddenstruturesforthis

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yieldsa linear-time reognition algorithm forthe lassof Helly irular-ar

graphs. Lin et al. introdued and haraterized the lass of proper Helly

irular-ar graphs[LSS07℄,those graphshavingairular-armodelwhih

issimultaneously properand Helly.

A irular-ar graph having a irular-ar model without two ars

ov-ering thewholeirle isallednormal irular-ar graph. Thisterminology

wasintroduedin[LS06b ℄. P.Hellprovedthatintervalbigraphsareexatly

thoseirular-argraphswithliqueoveringnumbertwoandhavinga

nor-mal irular-ar model [HH04 ℄. Generalizing irular-ar graphs, L. Alón

etal. introduedthe lassof loop graphs [ACH

+

07 ℄.

In spite of the fat that many researhers have been trying to nd the

listofforbiddensubgraphsthatharaterizesthelassofirular-argraph,

theproblemstill remainsopen. Inthis thesis we present somestepsinthis

diretion by providing haraterizations of irular-ar graphs by minimal

forbiddeninduedsubgraphs,when thegraphbelongsto anyof the

follow-ingfourdierentlasses: P

4

-free graphs,paw-free graphs,law-free hordal

graphs and diamond-free graphs. In addition, sineirular-ar graphs

be-longingtotheselasseshaveanormalirular-armodel,theseresultsimply

thatforbiddeninduedsubgraphsforthelassofnormalirular-argraphs

neessarilyontain adiamond, aninduedP

4

,an induedpaw andeither a

laworaholeas induedsubgraph. Wealso introdueand haraterize the

lassofsemiirulargraphs,irular-ar graphshavingairular-ar model

where its ars are semiirles. It is worth pointing out that all of these

lasses were studied along theway towards the proof of theStrong Perfet

GraphTheorem [Con89 , Ola88,PR76,Sei74 , Tu87℄. The aforementioned

resultshave beenpublished in[BDGS09℄.

A graph is dened to be irle ifit is the intersetion graph of a set C

ofhords on airle, suh aset is alledairle model. Cirlegraphs were

introduedbyEvenandItaiin[EI71 ℄tosolve anorderingproblemwiththe

minimumnumberofparallel stakswithouttherestritionofloading before

unloadingisompleted,provingthattheproblemanbetranslatedinto the

problemof nding the hromatinumber of a irle graph. Unfortunately,

this problemturnsoutto beNP-omplete [GJMP80℄.

Naji haraterized irle graphs in terms of the solvability of a

sys-tem of linear equations, yielding a O

(

n

7

)

reognition algorithm for this

lass[Naj85℄. The loal omplement of a graphG with respetto avertex

u 2 V

(

G

)

is the graph Gu that arises from G by replaing the indued

subgraph G

[

N

G

(

u

)]

by its omplement. Two graphs G and H are loally

equivalent ifand onlyifG arises fromH by anite sequene of loal

om-plementations. Bouhet proved that irle graphs are losed under loal

omplementation, aswell asthatagraphisirleifandonlyifeveryloally

(22)

bythisresult,GeelenandOum[GO09 ℄gaveanewharaterizationofirle

graphsintermsofpivoting. TheresultofpivotingagraphGwithrespetto

anedgeuvisthegraphGuv

=

Guvu(wherestandsforloal

om-plementation). AgraphG

0

ispivot-equivalent toGifG

0

arisesfromGbya

sequene of pivoting operations. They proved, with the aidof a omputer,

thatGisa irle graphifandonlyifeah graphthatispivot-equivalent to

Gontains none of15 presribed indued subgraphs.

Airlegraphwithairlemodelhavingallitshordsofthesamelength

isalledaunitirlegraph. Itiswell-knownthatthelassofproper

irular-argraphs is properlyontained inthelass ofirle graphs. Furthermore,

the lass of unit irular-ar graphs and the lass of unit irle graphs are

thesame [Dur00 ℄.

We say that G has a split deomposition if there exist two graphs G

1 and G 2 with jV

(

G i

)

j 3, i

=

1,2, suh that G

=

G 1 G 2 with respet

to some pair of marker verties (Ch. 3 of this thesis). If so, G

1

and G

2

arealled thefatorsof thesplit deomposition. Those graphs thatdo not

have a split deomposition are alled prime graphs. The onept of split

deomposition is dueto Cunningham[Cun82 ℄. Cirlegraphs turnedoutto

belosedunderthisdeomposition[Bou87℄andin1994Spinradpresenteda

quadrati-time reognizing algorithm forirle graphs thatbenetingfrom

this peuliarity [Spi94℄.

Cirle graphs are a superlass of permutation graphs. Indeed,

permu-tation graphs an be dened as those irle graphs having a irle model

suh that a hord an be added in suh a way that this hord meets all

thehordsbelonging to theirlemodel. Ontheother hand,permutations

graphs are those omparability graphs whose omplement graph is also a

omparability graph. Sine omparability graphs have been haraterized

byforbidden induedsubgraphs [Gal67℄, suh a haraterizations implies a

forbidden indued subgraphs haraterization for the lass of permutation

graphs.

Helly irle graphsarethose graphs having airle modelwhose hords

satisfy the Helly property; i.e, every set of pairwise adjaent hords have

a ommon point. This family of graphs was introdued by Durán [Dur00 ℄.

He also onjetured thata irle graphis Helly irle ifand onlyif itdoes

notontain a diamond as indued subgraph. Reently, this onjeture was

positivelysettled [DGR10 ℄. Nevertheless,Hellyirle graphs havenotbeen

haraterizedbyforbidden induedsubgraphs yet.

In this thesis we present some partial haraterizations by forbidden

indued subgraphs. We haraterize irle graphs among linear domino

graphs by proting from split deomposition. Consequently, we

harater-ize Helly irle graphs within the lass of lawfree graphs.

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tree-ographs and P 4

tidy graphs, are presented as an appliation of the

haraterization of Gallai for omparability graphs. Finally, we introdue

and haraterize the lass of unit Helly irle graphs, irle graphs having

a irle model whih is simultaneously Helly and unit. This results were

publishedin[BDGS℄.

LetG be ahereditary lassof graphs. A graphis denedto be probe G

ifits vertexset anbepartitioned into two sets: asetofprobeverties and

asetofnonprobeverties,sothatthesetofnonprobevertiesisastableset

and it an be obtained a graph belonging to G byadding edges with both

endpoints intheset ofnonprobe verties.

In1994 Zhagintrodued probe interval graphs as a researh toolin the

frameofthegenomeprojet,[ZSF

+

94℄. Sinethen,probeGgraphshavebeen

studiedfordierent hereditaryfamiliesofgraphsG. Shengharaterizedby

forbidden indued subgraphs those trees whih are probe interval [She99 ℄.

Brown et al. presented a haraterization byforbidden indued subgraphs

of probe unit interval graphs within thelass of trees [BLS09 ℄. Prºuljand

Corneil studied the forbidden subgraphs for probe interval graphs among

thelass of 2-treegraphs [PC05℄. Brown and Lundgrenproved that

bipar-tite probe interval graphs are equivalent to a the omplement of a lassof

irular-ar graphs whose lique number is two [BL06 ℄. In [BBd09 ℄, Bayer

etal. haraterize two sublasses ofprobe interval graphs, probe threshold

and probe trivially perfet graphs, in terms ofertain 2-SATformulas . In

thesameartile theypresent aharaterizationbyforbidden subgraphsfor

probe threshold graphs. Classes of probe G graphs, with G dierent from

intervalandunitintervalgraphs,havebeenalsostudiedformanyimportant

lassesofgraphs;e.g.,hordalgraphs[GL04,CGLS10℄,permutationgraphs

[CCK

+

09 ℄and split graphs [LdR07℄, amongothers.

In order to study the behavior of the join operation for probe interval

andprobeunitinterval graphs,weintroduetheonept ofhereditarylass

of graphs with a ompanion. We also present the whole list of all

forbid-den indued subgraphs whose omplement is disonneted for the lass of

probe interval graphs. In spite of we annot manage to do sofor the lass

of probe unit interval graphs, we present the list of forbidden subgraphs,

whose omplement is disonneted, for probe unit interval graphs, within

thelassesof tree-ographs andP

4

tidygraphs. We giveaharaterization

interms of forbidden indued subgraphsfor those o-bipartite graphs that

areprobe interval, this haraterization implies that o-bipartite probe

in-terval graphs ando-bipartite probeunit interval graphsarethesame lass

ofgraphs. Inaddition, probeinterval graphsandprobe unitintervalgraphs

areharaterized byforbidden subgraphs withinthelass oftree-ographs,

generalizing the haraterizations presented in [She99 ℄ and [BLS09 ℄,

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by forbidden indued subgraphs probe interval graphs and probe unit

in-tervalgraphs withinP

4

tidygraphs. We alsostudythelasses ofprobe

o-bipartite graphs and blok graphs, presenting a strutural haraterization

for probe o-bipartite graphs that leads to a polynomial-time reognizing

algorithmforthis lass. Probeblokgraphsareasublassofprobehordal

graphs, studied in [GL04, CGLS10℄. Probe hordal graphs have not been

haraterizedbyforbiddensubgraphsyet. InthisThesiswepresent a

har-aterizationforprobe blokgraphsbyforbidden induedsubgraphsandwe

prove that the lass of probe blok graphs is the intersetion between the

lasses ofhordal graphs and probediamond-free graphs [BDd

+

℄.

This Thesis is organized as follows. In Chapter 1 we give some

deni-tionsand abrief overview onthelasses westudied throughout this thesis.

Chapter2 isdevotedto partial haraterizations forirular-ar graphs. In

Chapter 3 we present partial haraterizations for irle graphs. Partial

haraterizations for probe interval graphs and probe unit interval graphs

arestudied inChapter 4 . Finally, inChapter 5 we present strutural

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(26)

Preliminaries

1.1 Denitions and notation

(

V 2 ,E 2

)

twographs. G 1 issaidto be isomor-phi to G 2

and vieversa, if there exists aone-to-one funtion f

:

V

1

!V

2

preservingtheadjaenies; i.e, vw2E

1 ifand onlyiff

(

v 1

)

f

(

v 2

)

2E 2 .

LetG

= (

V,E

)

beagraph. Thesetofvertiesadjaenttoavertexv2V

is alledneighborhood of v and denoted byN

jand is

alledthedegree ofv. Verties withdegree 0and jVj 1arealled isolated

vertex and universal vertex, respetively. A pendant vertex is a vertex

of degree one. H

= (

V

0 ,E

0

)

is said to be a subgraph of G if V

0 V and E 0 E. If, in addition, E 0

=

fuv 2 E

:

] =

N

[

w

]

. Let A,B V

(

G

)

. We say

thatAV is omplete to B V ifeveryvertex ofA isadjaent to every

(27)

LetG and H be two graphs. We saythat a graph Gdoes not ontain

H as indued subgraph ordoes notontain aninduedH ifanyindued

subgraph of G is not isomorphi to H. Given a olletion of graphs H , G

is dened to be H free if for any graph F 2 H , G does not ontain an

indued F. If H is a set with a single element H, we just use Hfree for

short. Thedisjoint union ofGandH isthegraphG[H whosevertexset

. Thedisjoint unionis

learlyan assoiative operation, and for eah nonnegative integer twe will

denote by tG the disjoint union of t opies of G. The join of G and H

is a graph G

+

H whose vertex set is V

(

G

)

[V

(

H

)

and whose edge set is

0

stands for the

graphwhosevertexset isV

(

G

)

andwhose edgeset isE

(

G

)

E

0 .

Givenalassofgraphs G,wedenote byo-G thelassofgraphs formed

by the omplements of graphs belonging to G. A lass of graph G is said

to be hereditary if for every indued subgraph G2 G, any subgraph of it

belongs to G. We saythat agraphGis non-G,ifGdoesnotbelong to the

lass G. If G is a hereditary lass, a graph G is dened to be minimally

non-G ifGdoesnotbelong to G buteveryproperindued subgraphdoes.

A stable set is a subset of pairwise non-adjaent verties. A omplete

set isasetofpairwiseadjaentverties. Aompletegraph isagraphwhose

vertex set is a omplete set. A lique is a omplete graph maximal under

inlusion. K

n

(n0) denotestheompletegraph onn verties. K

3 isalso

alled triangle. A diamond is the graph obtained from a ompleteK

4 by

removingexatlyone edge. A paw isthe graphobtainedfrom a triangle T

byaddingavertexadjaent toexatlyonevertexofT. Alique isasubset

of verties induing a omplete subgraph. A graph G is bipartite if V

(

G

)

an be partitioned into two stable sets V

1 , V

2

; G is omplete bipartite if

V 1 is omplete to V 2 . Denote by K r,s

the omplete bipartite graph with

jV 1

j

=

r andjV

2

j

=

s. A law is theomplete bipartitegraphK

1,3 .

Apath isalinearsequene ofdierentvertiesP

=

v

1 ,:::,v k suhthat v i isadjaent to v i

+

1 fori

=

1,:::,k 1. fv 2 ,:::,v n 1

garealled internal

verties ofthepath. Sumsinthisparagraph shouldbe understoodmodulo

k. If there is no any edge v

i v

j

suh that ji jj 2; i.e., all its internal

verties have degree two, P issaid to be either hordlees path or indued

path. DenotebyjPjthenumberofverties ofP. Ayleyle Cisalinear

sequene of verties C

=

v 1 ,:::,v k ,v 1 suh that v i is adjaent to v i

+

1 for

i

=

1,:::,k. If there is no any edge v i

v j

suh that ji jj 2, C is said

to be either hordlees yle or indued yle. By P

n

and C

n

we denote

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Figure 1.1: From left to right: law, diamond, gem,and 4-wheel

joiningtwonon-onseutive verties ofapathorayleinagraphisalled

ahord. A hole is anindued yle oflength at least 4.

Theoperationofedge subdivision inagraphGonsistsonseleting an

edgeuvofGandreplaingitwithaninduedpathu

=

v

1 ,v 2 :::,v k 1 ,v k

=

v with k 3 a positive integer. A prism is a graph that onsists of two

disjoint triangles fa 1 ,a 2 ,a 3 g and fb 1 ,b 2 ,b 3

g linked by three indued paths

P 1 ,P 2 ,P 3 whereP i links a i and b i fori

=

1,2,3.

A graph G is alled onneted if there is a path linking any two of its

verties. A maximal (under inlusion) onneted subgraph of G is alled

omponent of G. A graph G isantionneted if G is onneted; an

anti-omponentofGisthesubgraphofGinduedbythevertiesofaomponent

ofG.

LetGbea graph. v isalled aut vertex ifthenumberof omponents

of G v is greater than the number of omponents of G. G is said to be

2-onneted ifGisonnetedanddoesnothaveanyutvertex. Amaximal

2-onneted subgraphis alleda blok.

1.2 Overview on some lasses of graphs

Theaim ofthis setionis togive abrief overview ofsomelasses ofgraphs,

espeiallythoseweusethroughoutthisThesis. Themainfousisstrutural.

1.2.1 Domino graphs

A graph G is domino if eah vertex belongs to at most two liques. If, in

addition, eah ofits edgesbelongsto at mostone lique, thenG isa linear

dominograph.

Thefollowingtheoremgivesaforbiddeninduedsubgraph

harateriza-tion forthelassof dominoes graphs.

Theorem 1. [KKM95℄ G is a domino graph if and only if G is a

{gem,law,4wheel}free graph.

Notie that, given a graph G, then every edge belongs to at most one

liqueifandonlyifGisdiamondfree. Consequently,thefollowingorollary

(29)

c

d

a

b

e

a

b c

_d

e

Figure 1.2: Interval graphand its interval model

Corollary 1. [KKM95℄ G is a linear domino graph if and only if G is

a {law,diamond}free graph.

1.2.2 Intersetion graphs

GivenafamilyofsetsF,agraphG

= (

V,E

)

isdenedtobeanintersetion

graph respet on F if there is a one-to-one funtion f

:

V ! F, suh that

uv2E ifandonlyiff

(

u

)

andf

(

v

)

interset. Intersetiongraphshavebeen

widely studied in the literature. Having good strutural qualities, interval

graphs,hordalgraphs,irular-argraphsandirlegraphsaresomeofthe

most studied intersetion graph families. In this setion we will fous on

summarizingsomefeaturesofsomeoftheaforementioned lasses ofgraphs.

Interval graphs

A graph G

= (

V,E

)

is dened to be an interval graph if there exists a

family of open intervals I

=

fI

v g

v2V

in the real line and a one-to-one

funtionf

:

V !I suhthatuv 2E ifandonlyiff

(

u

)

andI

(

v

)

interset.

Suha familyof intervalsI isalled aninterval model of G.

Beforeintroduing thewell-known forbidden indued subgraph

hara-terizationforintervalgraphs,wewilldeneatoolthatplayaveryimportant

roleinthisharaterization. ThreevertiesinagraphGformanasteroidal

triple if,foreahtwo of threeverties,there existsapathontaining those

two butno neighborof thethird.

Boland and Lekerker haraterized interval graphbyforbidden indued

subgraphs. Theymanaged to do sohavingharaterized interval graphs as

those graphsnotontaining asteroidal triple [LB62℄.

Theorem 2. [LB62℄ A graph is an interval graph if and only if it

on-tains no indued bipartite-law, umbrella, n-net for any n 2, n-tent

for any n3,or C

n

for any n4.

Notie that allgraphs depited inFig. 1.3ontain anasteroidal triple.

A proper interval graph is an interval graph havingan interval model

suh that none of its intervals is properly ontained in another, suh an

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Figure 1.3: Minimal forbidden indued subgraphs for the lass of interval

graphs

interval graph having an interval model with all its intervals having the

same length, suh aninterval model isalled unitinterval model.

ProperintervalgraphswereintroduedbyRoberts[Rob69℄. Itwas

him-self thatharaterized those interval graphs thatareproperinterval.

Theorem 3 ([Rob69℄). Let G be an interval graph. G isunit interval if

and only if G does not ontain an indued law.

Wegner [Weg67℄ and Robert [Rob69℄ introdued unit interval graphs.

Notie that every unit interval graph is a proper interval graph. Roberts

wasabletoprovethattheonverseisalsotrue. Consequently,byombining

Theorem2 and Theorem3 follows thebelowtheorem.

Theorem4 ([Rob69℄). LetGbea graph. G isunitinterval graph ifand

only ifG does notontain an induedlaw, a net,a tent,or C

n

forany

n4.

Permutation graphs

In order to introdue permutation graphs we rst dene omparability

graphs. WesaidthatadigraphG

0

= (

V 0 ,E 0

)

isanorientationofagraphG ifV 0

=

V

(

G

)

2E 0 . Agraph

issaid tobe omparability ifithasatransitive orientation. Comparability

graphs were haraterized byGalai by meansofa list offorbidden indued

subgraphs [Gal67℄.

Theorem 5. [Gal67℄ A graph is a omparability graph if and only if it

does not ontain as an indued subgraph any graph in Figure 1.4 and

its omplement does not ontain as an indued subgraph any graph in

Figure 1.6.

Let

:

f1,:::,ng!f1,:::,ngbeapermutationofV

n

=

f1,:::,ng;i.e,

(31)

Figure 1.4: Some minimal forbidden indued subgraphs for omparability

graphs.

V n

andwhoseedgesetisformedbythoseunorderedpairsijsatisfyingi<j

and

1

(

i

)

> 1

(

j

)

. A graph Gis dened to be a permutation graph if

there existsa permutation suh that thegraph G

(

)

is isomorphito G.

Notie that ifyou plae {1,...,n} intwo parallel vertial opies of thereal

line and join iof theline on theleft with

(

i

)

intheline on theright, the

intersetion graph of these segments is isomorphi to G

(

)

. For instane,

onsider the permutation

:

V

4

! V

4

2, the graph G

[

]

is isomorphi to the intersetion

graph of the segments depited in Figure 1.5 . Pnueli et al. presented a

haraterizationofpermutation graphsthatshowstherelationshipbetween

this lassand omparability graphs.

1

2

3

4

1

2

3

4 a

b

c

d

a

b

d

c

Figure1.5: The intersetion graphof theset ofsegmentson the left whose

endpointsbelongtothevertiallinesisisomorphitothegraphontheright.

Theorem 6. [PLE71℄ A graph G is a permutation graph if and only if

G and G are omparability graphs.

Therefore, theharaterizationof omparability graphs in[Gal67℄leads

immediatelyto aforbidden induedsubgraph haraterization of

permuta-tiongraphs.

Corollary 2. A graph G is a permutation graph if and only if G and

G do not ontain as an indued subgraph any graph in Figure 1.4 and

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Figure1.6: Somegraphswhoseomplementsareminimalforbiddenindued

subgraphs foromparability graphs.

Chordal graphs an its sublasses

AgraphGisdenedto be hordal ifGdoesnotontainanyinduedyle

with at least four verties. Consequently, interval graphs are a sublass of

hordalgraphs. Furthermore,sinethegraphtentishordalbutnotinterval

(see,Fig.1.3 ),thelassofinterval graphs isproperlyontained inthelass

of hordal graphs. Chordal graphs were haraterized as the intersetion

graphof subtreesina tree [Gav74 ℄.

Split graphs arethose graphs whosevertex set an be partitioned into

two sets: a omplete set and an stable set. Split graphs are a sublass of

ographsandwereharaterizedbyforbiddeninduedsubgraphsasfollows.

Theorem 7. [HF77 ℄ Let G be a graph. G is a split graph if and only if

G does not ontain any indued 2K

2 , C 4 and C 5 .

Split omplete graphs are those split graphs suh that an be

parti-tioned into astable setand aompletesetin suha way thattheomplete

set is omplete to the stable set. Split omplete graphs will be also alled

probe omplete. The above theorem implies the following result whose

proof isstraightforward.

Lemma 1. Let G be a graph. G is split omplete if and only if G does

not ontain any indued C

4

and P

3 .

Gis ablok graph ifit isonnetedandeveryblokisaomplete. Itis

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Given two verties v and w of aonneted graph G, d

(

v,w

)

standsfor

thenumber of edges of a path with theminimum number of edges linking

v and w.

Aonneted graphisdenedto beptolemai ifandonlyifsatisesthe

ptolemaiinequality; i.e., foranyfourverties u,v,w and x

.

Howorkaprovedthatagraphisptolemaiifandonlyifitishordaland

gem-free [How81℄(see F

2

inFig. 5.2 ).

1.2.3 Cographs and its superlasses

Inthis setionwearegoingtooverviewthemostimportantstrutural

har-aterization of ograph and some of its superlasses. First, we will start

deningographsreursively.

A ograph anbe reursivelyonstrutedas follows:

1. The trivial graph isa ograph.

2. If G

1

and G

2

areographs, then G

1

[G

2

is aograph.

3. If Gis aograph,then Gis aograph.

Thereareseveralwaysofharaterizingographs(seee.g.,[CO00,CLS81,

CPS84 ,Sei74 ℄). Next,wegiveaharaterizationbyforibiddenindued

sub-graphforthis lass.

Theorem 8. [CLS81,CPS84 ℄ Let G

= (

V,E

)

a graph. G is a ograph if

and only if G is P

4 -free.

The following theorem shows a property of ographs whih is a useful

toolto dealwith this lass.

Theorem 9. [Sei74℄ If G is a ograph, then G is either not onneted

or not antionneted.

LetGbeagraphandletAbeavertexsetinduingaP

4

inG. Avertex

v of G is said a partner of A if G

[

A[fvg

]

ontains at least two indued

P 4

's. Finally, G is alled P

4

-tidy if eah vertex set A induing a P

4

in G

hasat mostone partner[GRT97℄.

ThelassofP

4

-tidygraphsisanextensionofthelassofographsandit

ontains manyother graph lasses denedbybounding thenumberof P

4 's

aordingtodierent riteria; e.g.,P

4

-sparse graphs[Hoà85℄,P

4

-litegraphs

[JO89 ℄,and P 4

(34)

A spider [Hoà85℄ is a graph whose vertex set an be partitioned into

three sets S, C, and R , where S

=

fs

1 ,:::,s k g (k 2) is a stable set; C

=

f 1 ,:::, k gisaompleteset;s i isadjaentto j ifandonlyifi

=

j (a thin spider),ors i isadjaent to j

ifand onlyifi6

=

j (a thik spider);R isallowed tobeemptyandifitisnot,thenallthevertiesinRareadjaent

to all theverties inC and nonadjaent to all the verties in S. Thetriple

(

S,C,R

)

is alled the spider partition. By thin

k

(

H

)

and thik

k

(

H

)

we

respetively denote the thin spider and the thik spider with jCj

=

k and

H the subgraph indued by R . If R is an empty set we denote them by

thin k

and thik

k

, respetively. Clearly, the omplementof a thin spider is

a thik spider, and vie versa. A fat spider is obtained from a spider by

adding a true or false twin of a vertex v 2 S[C. The following theorem

haraterizes thestrutureof P

4

-tidy graphs.

Theorem 10. [GRT97 ℄ Let G be a P

4

-tidy graph with at least two

ver-ties. Then, exatly one of the followingonditions holds:

1. G is disonneted. 2. G is disonneted. 3. G is isomorphi to P 5 , P 5 , C 5

, a spider, or a fat spider.

Let G be a graph and let A be a vertex set thatindues a P

4

in G. A

vertex v of G is said a partner of A if G

[

A[fvg

]

ontains at least two

induedP

4

's. Finally,GisalledP 4

-tidy ifeahvertexsetAthatinduesa

P 4

inGhasatmostonepartner[GRT97℄. ThelassofP

4

-tidygraphs isan

extension ofthelass of ographsand itontains manyother graphlasses

denedbyboundingthenumberofP

4

'saording to dierent riteria; e.g.,

P 4

-sparse graphs, P

4

-lite graphs [JO89 ℄, and P

4

-extendible graphs [JO91 ℄.

A spider is a graph whose vertex set an be partitioned into three sets S,

C,and R ,where S

=

fs 1 ,:::,s k g (k2) is a stableset; C

=

f 1 ,:::, k g is a omplete set; s i is adjaent to j

ifand only if i

=

j (a thin spider), ors

i

is adjaent to

j

ifand onlyif i6

=

j (a thik spider); R is allowed to

be empty and ifit is not, then all the verties inR areadjaent to all the

verties inC and nonadjaent to all the verties in S. The triple

(

S,C,R

)

is alled the spider partition. Clearly, the omplement of a thin spider is

a thik spider, and vie versa. A fat spider is obtained from a spider by

adding atrue orfalse twin ofa vertex v2S[C.

Tree-ographs [Tin89 ℄areanothergeneralization ofographs. Theyare

dened reursively as follows: trees are tree-ographs; the disjoint union

of tree-ographs is a tree-ograph; and the omplement of a tree-ograph

is also a tree-ograph. It is immediate from the denition that, if G is a

(35)

(36)

Cirular-ar graphs

2.1 Introdution

AgraphGisairular-ar(CA)graphifitistheintersetiongraphofaset

Sofopenarsonairle,i.e.,ifthereexistsaone-to-onefuntionf

:

V !S

interset. Suhafamilyofarsisalledairular-ar model(CA

model) of G. CAgraphs an bereognized in lineartime [MC03℄. Notie

thata graphisan interval graph ifit admits aCAmodelsuhthat theset

ofarsdoesnotovertheirle. Intervalgraphshavebeenharaterizedby

minimal forbidden induedsubgraphs [LB62℄ (see Chapter 1, Setion 1.2 ).

A graphG isa proper irular-ar (PCA) graph ifitadmits a CAmodel

inwhih no aris ontained inanother ar. Tuker gave a haraterization

ofPCA graphsbyminimal forbidden indued subgraphs[Tu74℄.

Further-more, this sublass an be reognized in linear time [DHH96 ℄. A graph

G is a unit irular-ar (UCA) graph if it admits a CA model in whih

all the ars have the same length. Tuker gave a haraterization by

min-imal forbidden indued subgraphs for this lass [Tu74℄. Reently, linear

and quadrati-time reognition algorithms for this lass have been shown

[LS06b ,DGM

+

06 ℄. Finally,thelassofCAgraphsthatareomplementsof

bipartitegraphswasharaterizedbyminimalforbiddeninduedsubgraphs

[TM76 ℄. T.Feder etal. haraterized thoseCA graphsthatareobipartite

byedgeasteroids[FHH99℄. Nevertheless,the problemof haraterizing the

whole lass of CA graphs by forbidden indued subgraphs remains open.

In this hapter we present some steps in this diretion by providing

har-aterizations of CA graphs by minimal forbiddenindued subgraphs when

the graph belongs to any of fourdierent lasses: P

4

-free graphs, paw-free

graphs, law-free hordal graphs and diamond-free graphs. All of these

lasses were studied along the way towards the proof of the Strong Perfet

GraphTheorem[Con89 ,Ola88,PR76,Sei74,Tu87℄. Theresultspresented

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2.2 Preliminaries

2.2.1 Denitions

Denote byG

thegraphobtained fromGbyaddingan isolated vertex. Ift

isa non-negative integer,then tGwill denotethe disjoint union oft opies

of G. A graph G is a multiple of another graph H if G an be obtained

from H by replaing eah vertex x of H by a non-empty omplete graph

M x

and adding all possible edges betweenM

x

andM

y

if and onlyifx and

y areadjaent in H.

LetGandH begraphs. Gisanaugmented H ifGisisomorphitoH or

ifGanbeobtainedfromH byrepeatedlyaddingauniversalvertex. Gisa

bloomed H ifthereexistsasubsetW V

(

G

)

suhthatG

[

W

]

isisomorphi

to H and V

(

G

)

W is either emptyor it indues inG a disjoint union of

non-empty omplete graphs B

1 ,B

2 ,:::,B

j

for some j 1, where eah B

i

is omplete to one vertex of G

[

W

]

, but antiomplete to any other vertex

ofG

[

W

]

. Ifeah vertex inW isomplete to at least one of B

1 ,B 2 ,:::,B j ,

we say that Gis afully bloomed H. The omplete graphs B

1 ,:::,B

j will

be referred as the blooms. Abloomis trivial ifit isomposed of onlyone

vertex.

2.2.2 Previous results

Speial graphs needed throughout this hapter are depited in Figure 2.1 .

For notational simpliity, in this hapter, we will use net and tent as

ab-breviationsfor2-net and 3-tent, respetively.

Bang-Jensen andHell proved thefollowing result.

Theorem11. [BH94℄LetGbeaonnetedgraphontaining noindued

law,net, C

4

, or C

5

. If G ontains a tent as indued subgraph, then G

isa multiple of a tent.

Theorem11allowstoprovidethefollowingdesriptionofalltheminimal

non-PCA graphs withinthelassof onneted hordal graphs.

Theorem 12. [BH94℄ Let G be a onneted hordal graph. Then, G is

PCA if and only if it ontains no indued law or net.

Reall that Lekkerkerker and Boland determined all the minimal

for-bidden indued subgraphs for the lass of interval graphs (f. Chapter 1 ,

Theorem2). This haraterization yields someminimal forbidden indued

subgraphs for the lass of CA graphs. Let H be a minimal forbidden

(38)

Figure 2.1: Some minimallynon-CA graphs.

thenH isminimallynon-CA. Otherwise,ifH isCA,thenH

isminimally

non-CA,and furthermore all non-onneted minimallynon-CA graphs are

obtainedthis way. Sine theumbrella, net,n-tent forall n3,andC

n for

all n 4 are CA, but the bipartite law and n-net for all n 3 are not,

this observation and Theorem2 leadto the following result.

Corollary3. [TM76℄Thefollowinggraphsareminimallynon-CAgraphs:

bipartite law, net

, n-net for all n 3, umbrella

, (n-tent) for all n3, and C n

for every n4. Any other minimally non-CA graph is

onneted.

Weallthegraphslisted inCorollary3basiminimallynon-CAgraphs.

Anyotherminimallynon-CAgraphwillbeallednon-basi. Thefollowing

result,whihgivesastruturalpropertyforallnon-basiminimallynon-CA

graphs,an be dedued fromTheorem 2and Corollary 3 .

Corollary 4. If Gis a non-basi minimally non-CA graph, then G has

an indued subgraph H that is isomorphi to an umbrella, a net, a

j-tent for some j 3, or C

j

for some j 4. In addition, eah vertex v

of G H is adjaent to at least one vertex of H.

Proof. Sine G is non-CA, in partiular, G is not an interval graph. By

Theorem 2 , G hasan indued subgraph H isomorphi to a bipartite law,

umbrella, j-net forj2, j-tent forj3, orC

j

forsomej 4. Sine Gis

non-basi minimally non-CA, H is isomorphi to umbrella, net, j-tent for

some j 3, orC

j

for some j 4. Moreover, sine G is not isomorphito

H

,everyvertex v of G H isadjaent to atleast one vertexof H.

Figure 2.1introdues thegraphs G

i

is non-empty beause G is minimally non-CA and it is

notisomorphito C j for j4. So, H N H

(

v

)

is non-empty and is neither a pathnor a hole, hene it

isnotonneted. LetQ

1 and Q 2 be two omponentsof H N H

(

v

)

. Then,

there are induedpaths P

1

and P

2

on H suh that the interior verties of

P i

areQ

i

,fori

=

1,2. Therefore,the following onditions hold:

1. eah of P

1

and P

2

hasat least three verties,

2. v isadjaent to noneof theinterior verties of P

1

and P

2

,and

3. v isadjaent to theendpoints of P

1

and theendpointsof P

2 . By denition,P 1 and P 2

have nointeriorverties inommon.

Suppose, by way of ontradition, that P

1

and P

2

have no ommon

endpoints. Let w be an interior vertex of P

1 , so w is antiomplete to the holeinduedbyfvg[V

(

P 2

)

on G. Then, fv,wg[V

(

P 2

)

induesaproper

subgraphof G (it is proper sine it does not ontain the endpoints of P

1 )

thatis notaCA graph,a ontradition.

Suppose next that P

1

and P

2

have exatly one endpoint in ommon.

Suppose,bywayof ontradition, thatP

1

hasatleast twointeriorverties.

Then,thereisaninteriorvertexwofP 1

thatisnon-adjaenttotheommon

endpoint ofP 1 andP 2 . Sinefwgisantiomplete to fvg[V

(

P 2

)

,fv,wg[ V

(

P 2

)

indues a proper subgraph in G (it is proper beause it does not

ontain the endpoint of P

1

that is not a vertex of P

2

) that is non-CA,

a ontradition. This ontradition proves that eah one of P

1

and P

2

hasexatly one interior vertex. Then, fvg[V

(

P

1

)

[V

(

P 2

)

would indue

on G a subgraph isomorphi to either G

3

or G

7

, both of whih are

non-CAgraphs. Sine G is minimally non-CA, V

(

G

) =

fvg[V

(

3

,aontradition.

Finally suppose that P

1

and P

2

have exatly two endpoints in

om-mon. Suppose,bywayofontradition, thatP

1

(40)

points. LetwbeaninteriorvertexofP 1

thatisadjaent tononeofits

end-points. Then, w is antiomplete to fvg[V

(

P

2

)

and thus fv,wg[V

(

P 2

)

indues a proper subgraph on G (it is proper beause it does not ontain

theneighbours ofw inH) thatisnon-CA,aontradition. This

ontradi-tion shows that eah one of P

1

and P

2

has at most two interior verties.

Thus, fvg[V

(

P 1

)

[V

(

P 2

)

indues on Geither K 2,3 ,G 2 orG 4 ,whih are

minimally non-CA graphs. Sine G isminimally non-CA,G is isomorphi

to one of them,a ontradition.

2.3 Partial haraterizations

2.3.1 Cographs

Results on ographs used throughout this subsetion an be foundin

Sub-setion 1.2.3.

Dene semiirulargraphs to betheintersetion graphs of open

semi-irleson a irle. Notie that semiirulargraphs areUCA graphs.

Theorem 14. Let G be a graph. The following onditions are

equiva-lent: 1. G is fP 4 ,3K 1 g-free. 2. G isan augmented multiple of tK 2

for somenon-negative integer

t.

3. G is a semiirular graph.

Proof.

(

1

)

(

2

)

Assume that G is a fP

4 ,3K

1

g-free graph. If G has less

than two verties, then G is a omplete (note that tK

2

with t

=

0 is an

emptygraph). So,we an assumethatGhasat least two verties. Sine G

is P 4

-free,byTheorem 9 ,Gis eithernotonneted ornotantionneted.

Sine Gis3K

1

-free, ifG isnotonneted, thenGhasexatlytwo

om-ponents. Moreover, both omponents are omplete graphs. Thus, G is a

multiple of K

2

. Suppose now that G is non-antionneted, and let H be

an antiomponent ofG. Sine H isfP

4 ,3K

1

g-free and antionneted, H is

either trivial or non-onneted and, in the seond ase, by the arguments

above H induesonGamultiple ofK

2

. Letsbethenumberof

antiompo-nentsofGthataretrivialandtbethenumberofantiomponentsofGthat

indue on Ga multiple of K

2

. Sine G isthejoin ofits antiomponents, G

is the join of a multiple of tK

2

and a omplete K

s

for some non-negative

integerstands. Consequently,GisanaugmentedmultipleoftK

2

forsome

non-negative integer t.

(

2

)

(

3

)

AssumethatGisanaugmentedmultipleoftK

2

forsome

non-negativet. Inpartiular,GisamultipleoftK 2

[sK

1

(41)

p

1 '

p

2 '

p

1 p

₂

q

1 '

q

1 a

b

c

d

e

a

c

d

b

Figure 2.2: The graph 2K

2

[K

1

and its semiirularmodel.

tand some s

=

0 or1. Inorder to prove that Gis a semiirulargraph, it

suesto provethat tK

2

[sK

1

isa semiirulargraph. Fixa irleC. Let

fp 1 ,p 0 1 g,:::,fp t ,p 0 t g,fq 1 ,q 0 1 g,:::,fq s ,q 0 s

g be t

+

s pairwise distint pairsof

antipodal points of C. For i

=

1,:::,t, let S 1 i

and S

2 i

be the two disjoint

open semiirles on C whose endpoints are p

i and p 0 i . For j

=

1,:::,s let T j

be an open semiirle on C whose endpoints are q

j and q 0 j . Then S 1 1 ,S 2 1 ,:::,S 1 t ,S 2 t ,T 1 ,:::,T s

isasemiirular modelfortK

2

[sK

1

(see Fig.

2.2 ).

(

3

)

(

1

)

We now prove that semiirular graphs are fP

4 ,3K

1 g-free

graphs. It is lear that 3K

1

is not a semiirular graph beause there is

notenough spaeonairle forthree pairwisedisjointsemiirles. We now

showthatP

4

isnotasemiirulargraph. Assume,bywayof ontradition,

thatthereisasemiirulargraphmodelforP

4 . LetV

(

P 4

) =

fv 1 ,v 2 ,v 3 ,v 4 g, wherev i isadjaent to v i

+

1

fori

=

1,2,3 and let S

=

fS 1 ,S 2 ,S 3 ,S 4 g be a

semiirularmodelforP

4

,wherethesemiirleS

i

orrespondstothevertex

v i . Sine v 1 and v 3 are non-adjaent, S 1 and S 3

are disjoint and have the

sameendpoints. Sinev

1

andv

4

arealsonon-adjaent,thesameholdsforS

1 andS 4 ,heneS 3

=

S 4

. Thisontradits thefat thatS

2 \S 3 isnon-empty butS 2 \S 4

isempty. ThisontraditionshowsthatP

4

isnotasemiirular

graph. Sine the lass of semiirular graphs is hereditary, a semiirular

graphis f3K

1 ,P

4 g-free.

Theorem15. LetGbe aographthatontainsaninduedC

4

, andsuh

thatall its proper indued subgraphs are CA graphs. Then, exatly one

of the following onditions holds:

1. G isisomorphi to K 2,3 or C 4 . 2. G isan augmented multiple of tK 2

(42)

Proof. Clearly, K 2,3

and C

4

are not augmented multiples of tK

2

, for any

integert2. AssumethatGisisomorphitoneitherC

4

norK

2,3

. Sineall

properinduedsubgraphs ofGareCA graphs,C

4

and K

2,3

arenotproper

indued subgraphs of G. We must prove that G is an augmented multiple

oftK

2

,forsomeinteger t2. LetH betheindued subgraphofG thatis

isomorphi to C 4 . Sine C 4

=

2K 2

, we maysuppose thatthere is avertex

v in G H. Sine C

4

is not an indued subgraph of G, v is adjaent to

at least one vertex of H. Sine G is fP

4 ,K

2,3

g-free, either v isadjaent to

threeverties ofH orv isompleteto H. Inasethatv isadjaent tothree

verties ofH wewill denotebyC

(

v

)

theinteriorvertexofthepathindued

by N

H

(

v

)

in H. Suppose there exists a vertex w of G H, w 6

=

v, that

is non-adjaent to v. If v were adjaent to three verties of H and w were

omplete to H, then the subgraph indued by fv,wg[V

(

H

)

in G would

ontain an indued P

4

, a ontradition. Thus, v and w are both adjaent

to three verties of H or they are both omplete to H. Next assume that

v and w are both adjaent to three verties of G. If C

(

v

) =

C

(

w

)

, then

fv,wg[

(

V

(

1

as indued subgraph.

As-sume, by way of ontradition, that there is an indued subgraph S of G

isomorphito 3K

1

. Clearly H andS have atmosttwo verties inommon.

If H and S had two verties in ommon, then the remaining vertex of S

would be a vertex of G H adjaent to at most two verties of H, a

on-tradition. If H and S had exatly one vertex in ommon, then the other

two verties of S would be adjaent to the same three verties of H. As

we notied above, this leads to a ontradition. We onlude that H and

S must have no verties in ommon. Let fv

1 ,v 2 ,v 3 g

=

V

(

S

)

. Sine the

verties of S are verties of G H and pairwise non-adjaent, all of them

areadjaent tothreeverties ofH orallofthemareompletetoH. Ifallof

themwereadjaent to threevertiesof H,thenC

(

v

themwereompleteto H,thenH[fv

1 ,v

2 ,v

3

ginduesinGagraphwhih

ontains an induedK

2,3

,a ontradition. We onludethatGis 3K

1 -free.

SineGisalso P

4

-free,byTheorem14 ,Gisanaugmentedmultipleof tK

2 .

Finally,sineGontains C

4

as an induedsubgraph, t2.

We an nowpresent themain haraterization of this subsetion.

Theorem 16. Let G be a ograph. Then, G is a CA graph if and only

if G ontains no indued K 2,3 or C 4 .