Efficient Grap h Representations This page intentionally left blank http://dx.doi.org/10.1090/fim/019 FIELDS INSTITUT E MONOGRAPHS THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S Efficient Grap h Representations Jeremy P . Spinra d American Mathematica l Societ y Providence, Rhod e Islan d The Field s Institut e for Researc h i n Mathematical Science s The Field s Institute i s named i n honour o f the Canadia n mathematicia n Joh n Charle s Fields (1863-1932) . Field s wa s a visionar y wh o receive d man y honour s fo r hi s scientifi c work, including election to the Royal Society o f Canada i n 190 9 and to the Royal Society of London i n 1913 . Amon g othe r accomplishment s i n the servic e o f the internationa l math - ematics community , Field s wa s responsibl e fo r establishin g th e world' s mos t prestigiou s prize fo r mathematic s research—th e Field s Medal . The Fields Institute fo r Researc h i n Mathematical Science s i s supported b y grants fro m the Ontario Ministry o f Education an d Trainin g and the Natural Science s and Engineerin g Research Counci l o f Canada . Th e Institut e i s sponsore d b y McMaste r University , th e University o f Toronto, the Universit y o f Waterloo, an d Yor k University , an d ha s affiliate d universities i n Ontari o an d acros s Canada . 2000 Mathematics Subject Classification. Primar y 05C62 , 05C17 , 05C50 , 05C85 , 05-00, 05-02, 68R10 , 68W01 , 68P05 , 68Q30, 68-01 . For additiona l informatio n an d update s o n this book , visi t www.ams.org/bookpages/fim-19 Library o f Congress Cataloging-in-Publicatio n Dat a Spinrad, Jerem y P. Efficient grap h representation s / Jerem y P . Spinrad . p. cm. — (Fields Institut e monograph s ; 19) Includes bibliographica l reference s an d index. ISBN 0-8218-2815- 0 (acid-fre e paper ) 1. Representations o f graphs. I . Title. II . Series. QA166.242.S65 200 3 200304510 6 511/.-dc21 CI P Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h as to copy a chapter fo r use in teachin g o r research . Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d the customary acknowledgmen t o f the source i s given. Republication, systemati c copying , or multiple reproduction o f any material in this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d be addressed to the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e made by e-mail to [email protected] . © 200 3 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l rights except thos e grante d to the United State s Government . Printed i n the United State s o f America. @ Th e paper use d i n this boo k i s acid-free an d falls withi n the guidelines established t o ensure permanenc e an d durability. This publicatio n wa s prepared b y The Fields Institute . Visit the AMS home pag e at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 0 8 07 06 05 04 03 Contents Explanatory Remark s 1 Chapter 1 . Introductio n 5 1.1. Grap h Theor y Backgroun d an d Terminolog y 6 1.2. Algorith m Backgroun d an d Terminolog y 6 1.3. Representatio n Backgroun d 8 1.4. Exampl e o f a Nic e Representation 1 1 1.5. Overvie w o f Problems i n Graph Representatio n 1 2 1.6. Exercise s 1 5 Chapter 2 . Implici t Representatio n 1 7 2.1. Implici t Representatio n an d Universa l Graph s 2 1 2.2. Generalize d Implici t Representatio n 2 2 2.3. Representation s wit h Ver y Shor t Label s 2 3 2.4. Distanc e Labelin g o f Graphs 2 5 2.5. Exercise s 2 7 Chapter 3 . Intersectio n an d Containmen t Representation s 3 1 3.1. Chordalit y an d Transitiv e Orientatio n 3 1 3.2. Technique s fo r Interva l Graph s 3 3 3.3. Generalization s o f Interval Graph s 3 4 3.4. Permutatio n Graph s an d Generalization s 3 8 3.5. Containmen t Representation s 4 1 3.6. Overla p Representation s 4 2 3.7. Generalize d Intersectio n Model s 4 4 3.8. Perfec t Graph s 4 6 3.9. Exercise s 4 7 Chapter 4 . Rea l Numbers i n Graph Representation s 5 3 4.1. Warren' s Theore m 5 4 4.2. Continuou s Nongeometri c Variable s 5 6 4.3. Decidabilit y Result s 5 7 4.4. Exercise s 5 7 Chapter 5 . Classe s Which us e Globa l Informatio n 5 9 5.1. Path s i n Trees 5 9 5.2. Chorda l Comparabilit y Graph s 6 0 5.3. Fill-i n Scheme s 6 5 5.4. Closur e Operation s 6 6 5.5. Weakl y Chorda l Graph s 6 7 vi Content s 5.6. Othe r Fill-i n Scheme s 6 8 5.7. Exercise s 6 8 Chapter 6 . Visibilit y Graph s 7 3 6.1. Countin g Visibilit y Graph s 7 4 6.2. Cliqu e Cove r Representatio n 7 4 6.3. Induce d Visibilit y Graph s 7 6 6.4. Optimizatio n o n Visibility Graph s 8 0 6.5. Lin e o f Sigh t Graph s an d Othe r Variant s 8 0 6.6. Exercise s 8 1 Chapter 7 . Intersectio n o f Graph Classe s 8 5 7.1. Som e Fundamental Propertie s 8 5 7.2. Intersection s o f Fundamental Propertie s 8 7 7.3. Weakl y Chorda l Comparabilit y Graph s 8 8 7.4. Othe r Generalize d Classe s 9 0 7.5. Observation s Regardin g AT-fre e co-AT-fre e Graph s 9 1 7.6. Ope n Problem s o n the Generalize d Classe s 9 4 7.7. Exercise s 9 5 Chapter 8 . Grap h Classe s Define d b y Forbidden Subgraph s 9 7 8.1. Cograph s 9 7 8.2. Classe s Which ar e too Larg e to hav e Efficien t Representation s 10 0 8.3. Relatio n Betwee n Recognitio n Problem s 10 5 8.4. Classe s define d b y Forbidding Set s o f Induced Subgraph s 10 5 8.5. Dilwort h Numbe r an d Pose t Widt h 10 7 8.6. Exercise s 10 9 Chapter 9 . Chorda l Bipartit e Graph s 11 1 9.1. T-fre e Matrice s 11 1 9.2. Countin g an d Representatio n 11 2 9.3. Characterization s 11 8 9.4. Recognitio n 12 0 9.5. Optimizatio n Problem s o n Chorda l Bipartit e Graph s 12 3 9.6. Variant s o f Chordal Bipartit e Graph s 12 5 9.7. Subclasse s o f Chordal Bipartit e Graph s 12 6 9.8. Perfec t Eliminatio n Bipartit e Graph s 13 0 9.9. Bipartit e Graph s wit h Forbidde n Induce d Subgraph s 13 1 9.10. Exercise s 13 2 Chapter 10 . Matrice s 13 5 10.1. Smal l Forbidden Classe s o f Matrices 13 5 10.2. Linea r Matrice s 13 6 10.3. Forbidde n 2 by 2 Identity Matrice s 13 8 10.4. Forbiddin g (™ ) 13 9 10.5. Othe r Classe s o f Interest 14 2 10.6. Th e Problem s o f Counting an d Representatio n 14 2 10.7. Othe r Matri x Propertie s 14 5 10.8. Exercise s 14 5 Chapter 11 . Decompositio n 149 Contents vi i 11.1. Substitutio n Decompositio n an d Verte x Partitioning 14 9 11.2. Joi n Decompositio n 16 0 11.3. Recursivel y Decomposabl e Graph s 16 3 11.4. Clique-widt h an d NLC-widt h 16 4 11.5. Cliqu e Separato r Decompositio n 16 7 11.6. Ske w Partition 17 0 11.7. 2-Joi n 17 4 11.8. Exercise s 17 5 Chapter 12 . Eliminatio n Scheme s 18 1 12.1. Distanc e Hereditar y Graph s 18 1 12.2. Strongl y Chorda l Graph s 18 2 12.3. k-Simplicia l Eliminatio n Scheme s 18 4 12.4.