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

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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. Doubl y an d Duall y Chorda l Graph s 18 5 12.5. Exercise s 18 7

Chapter 13 . Recognitio n Algorithm s 19 1 13.1. Chorda l Graph s 19 1 13.2. Interva l Graph s 19 3 13.3. Circular-Ar c Grap h Recognitio n 19 7 13.4. Restriction s o n Intervals an d Arc s 20 4 13.5. Trapezoi d Graph s an d Relate d Classe s 20 8 13.6. Circl e Graphs 21 2 13.7. Circula r Permutatio n Graph s 21 7 13.8. Weakl y Chorda l Graph s 21 8 13.9. Path s i n Tree s 21 9 13.10. NP-complet e Classe s 22 2 13.11. Ope n Classe s 22 4 13.12. Exercise s 22 4

Chapter 14 . Robus t Algorithm s fo r Optimizatio n Problem s 23 1 14.1. Robus t Algorithm s whic h ar e Faster Tha n Recognitio n 23 3 14.2. Problem s Helpe d b y a Representation 24 1 14.3. Ope n Problem s fo r Robus t Algorithm s 24 7 14.4. Complexit y Issue s 24 9 14.5. Exercise s 25 2

Chapter 15 . Characterizatio n an d Constructio n 25 7 15.1. Chorda l Graph s 25 7 15.2. Uni t Interva l Graph s 25 9 15.3. Uni t Circular-Ar c Graph s 26 0 15.4. Constructio n Proble m fo r NP-complet e Classe s 26 1 15.5. Exercise s 26 2

Chapter 16 . Application s 26 5 16.1. Computationa l Biolog y 26 5 16.2. Network s 26 8 16.3. Programmin g Language s 27 2 16.4. A Cautionary Tal e 27 2 16.5. Exercise s 27 3 viii Content s

Glossary 27 7 Survey o f Results o n Grap h Classe s 30 3 Bibliography 31 9 Index 33 7 This page intentionally left blank Glossary

I shoul d not e that thes e definition s sometime s ma y b e i n conflic t wit h genera l use o f a term . I f a ter m i s use d i n a specifi c manne r i n thi s book , an d thi s spe - cific usag e i s simple to explain , I choose to includ e th e definitio n whic h applie s t o this boo k rathe r tha n a mor e genera l definitio n whic h migh t b e harde r t o under - stand. Fo r example , thi s boo k doe s no t discus s multigraphs , an d i n som e case s the definitio n I choos e to includ e woul d no t correspon d t o standard definition s o n multigraphs. I also do not worry about the fact that m y definition o f order notatio n symbols i s nonstandard i f negative function s ar e allowed , sinc e these ar e use d onl y for tim e analyse s i n this book .

Adjacent Verte x x i s adjacent t o vertex y i f there i s an edg e fro m x t o y.

Adjacency lists: A form o f graph representatio n i n which neighbor s o f vertex i ar e stored i n a linke d list . I n general , thi s lis t i s assume d t o b e unordered , i n th e sense that verte x j ma y preced e verte x k o n a n adjacenc y lis t eve n though j < k. However, i t i s possible to sort al l adjacency list s in 0(n+ra) time , i f this i s desired. Technically, a n adjacenc y lis t i s a matri x o f linke d lists , thu s allowin g th e lis t o f neighbors o f vertex i to b e accesse d i n constant time .

Adjacency matrix: A form o f graph representatio n i n which a graph i s stored a s a n n x n matrix . Th e entr y (i , j) o f the adjacenc y matri x i s 1 i f i ha s a n edg e t o j, and 0 otherwise.

Anti-hole: A hole (chordles s induce d cycl e o f length a t leas t 4 ) i n the complemen t graph.

Arboricity: Th e arboricit y o f G i s the minimu m numbe r o f forests suc h that ever y edge o f G i s contained i n one o f these forests .

Arc: A connected subse t o f a circle . Als o use d a s a n alternat e nam e fo r edge , es - pecially i n a tree.

A steroidal triple: A trio o f mutually nonadjacen t vertice s x,y yz wit h th e propert y that fo r each pair o f vertices in the trio, there is a path between these vertices which avoids all neighbors o f the third vertex in the trio. Graph s without asteroida l triple s are als o calle d AT-free .

277 278 Glossary

AT-free: A graph i s AT-free i f it contain s n o asteroidal triples .

Augmented adjacency matrix: Entrie s ar e th e sam e a s i n th e adjacenc y matrix , except wit h I s alon g the mai n diagona l rather tha n Os .

Autonomous set: Alternativ e nam e fo r module .

Balanced k-module: A subset S o f vertices induce s a balanced /c-modul e i f there i s a partitio n Si , S2 o f S suc h that |S|/ 3 < |Si | < 2|S|/3 , an d S i ca n b e partitione d into at mos t k subsets Si ^ i n such a way that ever y vertex o f S 2 i s either adjacen t to ever y vertex o f Si^, o r to n o vertex i n S^ .

Bandwidth: Th e bandwidt h o f a n orderin g o f th e vertice s o f G i s th e maximu m difference betwee n endpoin t position s o f an y edge . Th e NP-complet e bandwidt h problem ask s whether ther e i s any ordering o f the vertices which has bandwidth a t most k.

Berge graph: A grap h withou t an y od d hole s o r od d anti-hole s o f lengt h greate r than three ; thes e hav e recentl y bee n show n to b e equivalen t t o perfec t graphs , re - solving a famou s ope n question .

Biconvex graph: Bipartite graph with the property that ther e ar e orderings o f each color class satisfying fo r all x, the neighbors o f x occur consecutively in the ordering.

Bipartite adjacency matrix: Matri x representatio n o f a bipartite graph . Row s cor - respond t o vertice s fro m on e colo r class , column s t o vertice s fro m th e othe r colo r class, with a 1 in row i column j i f vertex i i s adjacent t o vertex j, an d 0 otherwise.

Bipartite graph: A grap h wit h th e propert y tha t vertice s ca n b e partitione d int o sets X, Y suc h that ever y edg e goes between a vertex i n X an d a vertex i n Y. Th e sets X an d Y ar e calle d colo r classe s o f G.

Bisimplicial edge: An edge (x, y) o f a bipartite graph such that N(#) U N(y) induce s a complet e bipartite graph .

Bit: A single digi t i n a binary (0/1 ) representatio n o f an object .

Block: I n a partition o f a set, eac h subse t i s called a block .

Bounded substitution diameter decomposition: A grap h ha s substitutio n decom - position diamete r k i f ever y prim e nod e i n th e decompositio n tre e ha s a t mos t k children. I f a clas s o f graph s ha s substitutio n decompositio n diamete r a t mos t k for k fixed , the n the clas s has bounded substitutio n decompositio n diameter .

Bounded tolerance graph: Toleranc e grap h suc h tha t n o toleranc e valu e i s large r than th e lengt h o f th e interval . Thi s implie s tha t i f a n interva l correspondin g t o vertex x contain s the interva l correspondin g t o y, th e vertice s mus t b e adjacent . Glossary 279

Boxicity: Th e minimu m dimensio n d suc h tha t a grap h ca n b e represente d a s a n intersection grap h o f c/-dimensiona l rectangles , wit h al l rectangle s oriente d alon g the axes . Boxicit y 1 graphs correspon d t o interva l graphs .

Brittle graph: A grap h G i s brittle i f there i s a n eliminatio n schem e fo r G whic h successively remove s eithe r a vertex whic h i s not a midpoin t o f an y P4 , o r no t a n endpoint o f any P4 .

Ci'. A chordless cycl e on i vertices.

Certificate: A certificate o f a property i s a proo f tha t th e propert y holds . Certifi - cates ar e important i n the theory o f NP-completeness; N P ca n be define d i n term s of existenc e o f polynomia l siz e certificates . Certificate s ar e use d i n a numbe r o f ways i n this book ; i n algorithms , th e existenc e o f a certificat e o f siz e f(n) ) ca n b e a separat e proble m fro m finding a n 0(f(n ) algorith m t o solv e th e problem , an d our discussio n o f robust algorithm s deal s wit h issue s whic h aris e i f w e are give n a certificate that a graph is in a class as opposed to when such a certificate i s not given.

Chain graph: A bipartite graph with the property that fo r eac h pair o f vertices u, v from th e sam e colo r class , either ~N(u) C N(^) o r N(v ) C N(u) .

Child: In a rooted tree , children o f v ar e neighbors o f v other than the parent o f v.

Chord: In a cycle , a n edg e whic h goe s between vertice s whic h ar e no t consecutiv e on the cycle . O n a circle , a straight lin e connecting tw o points o n the circle .

Chordal graph: A graph wit h n o induced (i.e . chordless ) cycl e o f length > 3 .

Chordal bipartite graph: A bipartite graph with no chordless cycles of length greater than 4 . A n important characterizatio n i s that G i s chordal bipartit e i f and onl y i f the bipartit e adjacenc y matri x ha s a T-fre e ordering .

Chromatic number: Th e minimu m numbe r o f colors necessar y to colo r a graph .

Circle graph: Intersection grap h o f chords o f a circle .

Circle order: Containment grap h o f disks i n the plane .

Circuit: Alternativ e nam e fo r cycle .

Circular-arc graph: Intersection grap h o f arcs o f a circle.

Circular permutation graph: Intersection graph s o f curves connecting points on two concentric circle s o f differen t diameter , suc h tha t n o pai r o f curve s intersect s a t more than on e point .

Circular Is property: A circular orderin g o f a set, whic h obey s a se t o f restriction s requiring particula r subset s to appea r consecutivel y i n the circula r order . 280 Glossary

Classically NP-complete: A proble m P i s classicall y NP-complet e o n a restricte d domain D o f input s (suc h a s a clas s o f graphs ) i f ever y proble m i n N P ca n b e mapped i n polynomia l tim e an d spac e t o a n instanc e o f P fro m Z) , such tha t th e answer to the original problem is the same as the answer to the instance mapped to.

Claw-free graph: A graph wit h n o induced subgrap h equa l to K\$.

Clique: A set o f mutually adjacen t vertices .

Clique cutset: A cutse t whic h i s also a clique .

Clique separator: Alternat e nam e fo r cliqu e cutset .

Clique separator decomposition: A for m o f grap h decompositio n whic h take s a n arbitrary cliqu e cutse t C , decompose s th e grap h int o S L) C fo r eac h connecte d component S o f G — C, an d decompose s eac h subgraph recursively .

Clique problem: I n decisio n form , th e proble m o f whether a n inpu t grap h G has a clique o f siz e equa l t o inpu t numbe r k. Als o use d i n optimizatio n for m t o denot e the problem o f determining the cardinality o f the maximum cliqu e in a graph. Th e decision proble m i s NP-complete .

Clique tree representation: A n intersectio n mode l fo r a chorda l grap h G. Ther e i s a tre e T , suc h eac h nod e o f T correspond s t o a maxima l cliqu e o f G , an d clique s containing eac h individual verte x v correspon d t o a connected subtre e o f T. G ha s a cliqu e tree representation i f and onl y i f G i s chordal, an d thi s representation ca n be foun d i n linear time .

Clique-width: Th e minimu m numbe r o f label s neede d t o construc t a grap h fro m one vertex graphs using the operations union, addition o f all edges between vertice s with label i and vertices with label j, an d the relabeling operation which gives label j to al l vertices with curren t labe l i.

Clone: A clon e c of vertex v with respect to a set o f vertices S i s a vertex such tha t N(c) D S - v = N(v) OS -v.

Closure: Th e ^-closur e o f a grap h i s forme d b y repeatedl y addin g edge s betwee n nonadjacent pair s x , y o f vertice s suc h tha t degree(x ) + degree(y) i s a t leas t £; , until no further edge s can be added. I n a variant calle d fc'-closure, edges are adde d between nonadjacen t x,y suc h that \N(x) U N(y)\ i s at leas t k.

Co-: Fo r a graph clas s C, a graph i s a co-C graph i f the complemen t i s in C.

Co-NP: A decisio n proble m P i s i n co-N P i f the proble m o f decidin g whethe r th e answer t o P i s no i s in NP; i.e. , n o answer s hav e a polynomial siz e certificate . Glossary 281

Cograph: In th e origina l definition , a grap h whic h ca n b e buil t fro m singl e verte x graphs usin g the operations o f complement an d union . A n important characteriza - tion i s that thes e ar e exactly the graphs withou t an y P4 .

Coloring: A n assignmen t o f number s (calle d colors ) t o vertice s o f a graph , suc h that n o edg e connect s vertice s wit h th e sam e color . Th e questio n o f whethe r i t is possibl e t o colo r a grap h wit h k color s i s NP-complet e eve n fo r k = 3 . I n th e optimization versio n o f the colorin g problem, th e goa l i s to colo r a graph wit h th e minimum numbe r o f different colors .

Color class: I n thi s book , thi s refer s t o on e o f the subset s o f vertices use d t o par - tition a bipartite graph . Mor e generally , a colo r clas s i s a se t o f vertices give n th e same colo r b y a coloring assignment .

Comparability graph: An undirecte d grap h suc h that al l edges can b e assigne d di - rections, suc h that i n the directe d grap h wheneve r a —> b and b — * c , then a —> • c.

Complement: Th e complemen t o f G , writte n a s G , ha s th e sam e verte x se t a s G , and a n edg e fro m x t o y i n G i f and onl y i f there i s no edge fro m x t o y i n G .

Complete graph: Alternative nam e fo r clique .

Complete bipartite graph. A bipartite grap h whic h ha s a n edg e betwee n eac h pai r of vertices fro m differen t colo r classes .

Completely k-decomposable: A graph i s completely /c-decomposabl e with respect t o a decompositio n i f ever y induce d subgrap h containin g mor e tha n k vertice s k i s decomposable.

Completion problem: Th e completio n proble m fo r a clas s o f graphs C ask s fo r th e minimum numbe r o f edges necessary t o add t o a n input grap h G , s o that th e aug - mented grap h i s in G .

Composition sequence: A clas s o f graph s ha s a compositio n sequenc e i f ther e i s a countabl e sequenc e Gi,..,G;,.. . o f graph s i n th e clas s suc h tha t eac h Gi i s a n induced subgrap h o f G^+i , an d ever y grap h i n the clas s i s a subgraph o f som e Gi in the sequence .

Connected component: Maxima l connecte d subgraph .

Connected graph: A grap h suc h that ther e i s a path betwee n eac h pai r o f vertices.

Connected matrix: A matri x suc h that n o pair o f rows and column s induce s 12-

Consecutive Is property: A n orderin g o f a se t whic h obey s restriction s requirin g elements i n various subsets t o occu r consecutively .

Construction problem: Th e problem o f constructing a form o f graph representation . 282 Glossary

Containment graph: Th e containmen t grap h o f a se t o f object s ha s a verte x cor - responding t o eac h objects , an d a directe d edg e fro m x t o y i f objec t x contain s object y.

Containment representation: A representation o f a graph a s a containment graph .

Contraction: Th e grap h forme d b y contractio n o f a n edg e (x , y) replace s th e pai r of vertices x , y b y a new vertex z , with N(z ) = N(x) U N(y).

Convex graph: Bipartite grap h suc h that vertice s o f one colo r clas s can be ordere d so that fo r ever y verte x v fro m th e othe r colo r class , N(v ) occur s consecutivel y i n the ordering .

Convex fan: A polygon P i n whic h ther e i s a singl e endpoin t v wit h th e propert y that ever y other endpoint w o f P i s either a neighbor o f v on P, o r the line segment connecting v an d w i s entirely insid e o f P.

Convex vertex: Give n a mode l fo r a visibilit y graph , a vertex i s conve x i f i t map s to a n endpoin t correspondin g t o a polygon angl e o f less than 18 0 degrees.

Cook's theorem: Th e theore m provin g th e NP-completenes s o f CNF-satisfiability . Almost al l NP-completeness proof s eventuall y deriv e fro m Cook' s theorem .

Cotree: A representation o f a cograph. Vertice s correspond t o leave s o f a tree, an d internal node s o f th e tre e ar e labele d wit h 0 an d 1 . Tw o vertice s u,v o f G ar e adjacent i f and onl y i f the least commo n ancesto r o f u and v in the cotree has labe l 1. G can be represented b y a cotree i f and onl y i f G i s a cograph .

Counting problem: Th e proble m o f determining th e numbe r o f graphs i n a class.

Covering graph: Th e undirecte d grap h forme d b y removin g direction s fro m th e transitive reductio n o f a partial order .

Cowly perfect: Litera l translatio n o f on e o f my favorit e name s fo r a grap h class , vachement parfait , whic h unfortunatel y di d no t find a plac e i n thi s book . Thos e interested i n correc t name s fo r grap h classe s coul d debat e whethe r o r no t suc h a class should b e bull-free .

Cutset: A set S o f vertices suc h that G - S i s disconnected .

Cycle: A sequenc e o f vertices i>i , i>2,..., Vf c such that fo r i i n the rang e l../c-l , Vi is adjacent t o ^+i, an d Vk is adjacent t o v\.

Cycle-free partial order: A partial orde r suc h that th e underlyin g graph i s chordal.

Decision problem: A proble m suc h tha t th e answe r i s either ye s o r no . Th e ter m NP-complete applie s b y definitio n t o decisio n problems . Glossary 283

Degree: Fo r th e purpose s o f thi s boo k (i n whic h self-loop s ar e neve r used) , th e number o f edges out o f a vertex .

Deletion problem: Th e deletio n proble m fo r a clas s C o f graphs ask s fo r th e min - imum numbe r o f edge s necessar y t o delet e fro m a n inpu t grap h G s o tha t th e resulting grap h i s in C.

Dense: Havin g man y edges . Ofte n use d i n a n informa l sense , but sometime s use d to mea n numbe r o f edges i s ©(n 2).

Descendant: A vertex d i n a rooted tre e i s a descendan t o f v i f the pat h fro m th e root t o d goes through v.

Deterministically k-decomposable: A graph i s deterministically decomposabl e wit h respect t o a decompositio n i f fo r ever y possibl e choic e o f recursiv e decompositio n steps, ever y prim e componen t ha s siz e at mos t k.

Diameter. Th e diamete r o f a graph i s the maximu m distanc e betwee n an y pai r o f vertices.

Digraph: Directe d graph .

Dilworth Number: Th e maximu m cardinalit y o f a set S o f vertices i n a graph suc h that fo r al l pairs o f vertices x , y i n 5, som e neighbo r o f x i s not i n N[y] and som e neighbor o f y i s not i n N[x].

Dimension: Th e minimum number o f linear extensions o f a partial order which give the partial order a s their intersection. I n other words, the minimum number o f lists of vertices o f a partial order P suc h that x < y i f and only i f x precede s y in all lists.

Disk intersection graph: Intersection grap h o f disk s i n the plane , wher e a dis k de - notes a circl e plu s the interio r o f a circle ; thus, i f on e dis k contain s th e other , th e corresponding vertice s ar e adjacent .

Distance: Th e distance betwee n x an d y i s the lengt h o f the shortest bat h betwee n x an d y.

Distance hereditary graph: A grap h suc h tha t fo r ever y pai r o f vertice s x , y, al l chordless paths fro m x to y hav e the same length. Relevanc e to this book come s in part fro m a n elimination schem e characterization; a connected grap h G i s distance hereditary i f and only i f it can be reduced to a single vertex by repeated eliminatio n of twins o r pendant vertices .

Distance labeling: Generalizatio n o f implici t representatio n t o th e distanc e func - tion. Eac h verte x i s assigne d a label , an d th e distanc e betwee n x an d y ca n b e computed usin g only the labe l informatio n store d a t x an d y. 284 Glossary

Dominating pair. A pai r o f vertice s x , y i s a dominatin g pai r i f fo r al l vertice s z , every path fro m x t o y contain s a t leas t on e neighbor o f z.

Dominating set: A set S o f vertices suc h that ever y verte x o f V - S i s adjacent t o at leas t on e verte x i n S. Decidin g whethe r a grap h ha s a dominatin g se t o f siz e input numbe r k i s NP-complete. I n the optimization version , i t i s necessary to find the smalles t cardinalit y dominatin g set .

Domination graph: A grap h G with the propert y tha t fo r ever y induce d subgrap h H o f G , there i s a pair o f vertices x, y suc h that N(x) C N[y ] i n H.

Dot-product representation: Representatio n o f a grap h i n whic h eac h verte x i s as- sociated wit h a vector , an d vertice s ar e adjacen t i f and onl y i f the dot-produc t o f the corresponding vector s i s at leas t 1 . Th e dot-product dimensio n i s the minimu m number o f element s i n vector s whic h ca n b e use d t o represen t G. Representin g graphs wit h constan t dot-produc t dimensio n implicitl y i s open.

Doubly chordal graph: A grap h i s doubly chorda l i f there i s an orderin g o f the ver - tices v\V2-.v n suc h that eac h vi i s simplicial i n the graph induce d b y Vi..v n, an d Vi has a maximal neighbo r i n the grap h induce d b y Vi+i..v n.

Doubly convex graph: Alternate nam e fo r biconve x graph .

Dually chordal graph: Grap h whic h ca n b e ordere d v\..v n suc h that eac h Vi has a maximal neighbo r i n the grap h induce d b y Vi..v n.

Edge: A connectio n between two vertices. Thi s can be written as an unordered pai r or a s u - v fo r a n undirected graph , an d a s an ordere d pai r o r u— > v fo r a directe d graph.

Efficient representation: Thi s i s a n informa l ter m use d throughou t th e book . I n general, i t mean s tha t thi s for m o f representatio n ha s importan t advantage s ove r standard methods fo r representing a class of graphs, even though the representatio n might no t b e space optimal, an d adjacenc y testin g might no t take constant tim e o r be distributed t o bits store d a t vertices .

Elimination scheme: A n orderin g v\V2-.v n o f vertice s o f a graph , suc h tha t eac h vertex Vi satisfies som e particular propert y P i n the grap h induce d b y Vi..v n.

Envelope: Th e envelop e o f a set o f lin e segment s consist s o f the unio n o f line sub- segments o n the infinit e fac e o f the unio n o f the segments .

EPT graph: Intersection grap h o f paths i n a tree, where w e say two paths intersec t if they shar e a common edg e (i.e . tw o paths whic h intersec t onl y a t a vertex mus t correspond t o nonadjacen t vertices) . Glossary 285

External visibility graph: A grap h i n whic h vertice s correspon d t o endpoint s o f a polygon, an d tw o vertice s ar e adjacen t i f the lin e segmen t connectin g the m i s en - tirely outsid e the polygon . f-diagram: Structur e use d i n a representation o f co-comparability graphs . Vertice s correspond t o curves connecting points o n two parallel lines , with curve s intersect - ing (a t leas t once ) i f an d onl y i f th e correspondin g vertice s ar e adjacent . Thi s representation i s possible i f and onl y i f G i s a co-comparability graph .

Feedback vertex set: A set S i s a feedbac k verte x se t i f G - S i s acyclic. Th e feed - back verte x se t proble m ask s fo r the minimu m cardinalit y feedbac k verte x set ; th e decision versio n o f feedback verte x se t i s NP-complete .

Ferrers digraph: A directed grap h suc h that fo r al l pair s x , y, eithe r N[x ] contain s N(y) o r N[y] contains N(x).

Ferrers dimension: Th e Ferrer s dimensio n o f a directe d grap h G i s the minimu m number o f Ferrers digraphs whic h yiel d G a s their intersection .

Fill-in scheme: A method fo r constructin g a graph b y repeatedl y addin g edge s t o the curren t grap h usin g som e se t o f rules.

Generalized implicit graph question: Th e open questio n a s to whether ever y hered - itary clas s o f graphs with 2^ n) member s has a generalized implici t representation .

Generalized implicit representation: Fo r a graph clas s with 2^ n^ graph s o n n ver - tices, a representation whic h store s 0(f(n)/n) bit s a t eac h vertex , suc h that adja - cency betwee n x an d y ca n be tested usin g onl y the bit s stored a t x an d y.

Global information: Informatio n representin g a graph i s global i f it can be accesse d from mor e than on e specifi c verte x o f the graph .

Graph: An ordered pai r (V , E), wher e V i s a set o f vertices and E i s a set o f edges. Edges ar e unordere d pair s o f vertices.

Graph class: A (usuall y infinite ) se t o f graphs.

Graph isomorphism: Th e proble m o f determining whethe r tw o graphs ar e isomor - phic. N o polynomial algorithm fo r graph isomorphism i s known, and graph isomor - phism i s not know n to b e NP-complete .

Greedy algorithm: A n algorithm whic h add s to its current solutio n sequentiall y ac - cording t o som e criterion , an d neve r backtrack s b y removin g a n elemen t fro m it s current solution .

Greedy coloring: The assignmen t o f colors to vertices accordin g to a n order , whic h assigns th e nex t verte x v th e smalles t colo r numbe r whic h ha s no t alread y bee n assigned to a neighbor o f v. I n the context o f circular-arc graph coloring , the orde r 286 Glossary corresponds t o nex t ar c i n the circular-ar c model .

Grid: Grid i s used i n tw o differenc e sense s i n this book . I f a geometrically define d class o f graphs i s defined, placin g the model on a grid means assigning integer coor - dinates to all objects used in the definition. Th e grid is also a special graph in which endpoints hav e label s (b,c) fo r 6 , c in som e range s l..i,l.. j an d tw o vertice s (6 , c) and (d , e) ar e adjacen t i f and onl y i f b = d and c differs fro m e by 1 , or c = e an d b differs fro m d by 1 . A graph i s a grid graph i f it i s an induced subgraph o f the grid.

Grid intersection graph. Intersectio n grap h o f a se t o f lin e segment s i n the plane , where eac h lin e segment i s either horizonta l o r vertical .

Hamilton cycle: A Hamilton cycl e i n a grap h i s a cycl e whic h goe s through ever y vertex exactly once. Als o called Hamilton circut. Th e problem o f deciding whethe r a grap h ha s a Hamilto n cycl e i s NP-complete, a s i s deciding whethe r a graph ha s a similarl y define d Hamilto n path .

Height: I n this book, the height o f a poset i s the length o f a longest path (o r size of a maximu m chain ) i n the poset . Thus , w e would sa y a bipartite pose t ha s heigh t 1, while this i s called heigh t 2 in som e texts.

Helly property: A set ha s the Hell y property i f for al l subsets S suc h that eac h pair of object s i n S ha s a commo n intersection , the n th e entir e se t S ha s a t leas t on e common intersectio n point . Fo r intersectio n representation s o f graphs, thi s mean s that al l cliques share a common poin t i n the representation .

Hereditary: A clas s o f graph s i s hereditar y i f fo r ever y G i n th e clas s an d ever y vertex v o f G , G - v i s also i n the class .

Hole: Chordles s cycl e o f lengt h a t leas t 4 . Thi s boo k allow s hole s t o b e o f eve n length, whil e som e papers reserv e the nam e fo r od d lengt h chordles s cycles .

Homogeneous Pair. Verte x set s A,B suc h tha t |A | + |JB | > 2 , A i s a modul e o f G — B, an d B i s a module o f G — A. Als o known a s a 2-module .

Homogeneous set: Alternativ e nam e fo r module .

House: Th e specifi c grap h o n 5 vertices consistin g o f a cycl e o n 4 vertices, an d a 5th vertex adjacen t t o 2 neighbors i n the cycle .

Hypergraph: A hypergraph differ s fro m a graph i n that eac h edg e (calle d a hyper - edge) i s a subset o f any number o f vertices, rather tha n bein g a subset o f exactly 2 vertices.

Identity matrix: A square matrix wit h I s on the mai n diagonal , an d O s everywhere else. Glossary 287

Immediate predecessor/successor. I n a partial order , thes e ar e vertice s whic h hav e edges to/from v i n the transitive reduction .

Implicit graph conjecture: Th e conjectur e tha t ever y hereditar y clas s o f graph s which ha s 2°(nlogn) graph s o n n vertice s mus t hav e a n implici t representation .

Implicit representation: Grap h representation whic h stores O(logn) bits per vertex , such that adjacenc y betwee n x an d y ca n be tested usin g onl y the bit s store d a t x and y.

Indegree: The numbe r o f edges into a vertex o f a directed graph .

Independent set: A se t o f mutuall y nonadjacen t vertices . Determinin g whethe r a graph ha s a n independen t se t o f siz e inpu t paramete r k i s NP-complete . I n th e optimization problem , yo u as k fo r th e maximu m cardinalit y independen t set .

Induced subgraph: The subgraph induce d b y a set S o f vertices i n a graph G has S as its vertex set , with tw o vertices adjacen t i n the induce d subgrap h i f and onl y i f these vertices ar e adjacen t i n G.

Induced visibility graph: A grap h whic h i s a n induce d subgrap h o f som e visibilit y graph.

Intersection class: A graph clas s whic h correspond s exactl y t o intersectio n graph s of a particular typ e o f object .

Intersection graph: Th e intersectio n grap h o f a se t o f object s ha s a verte x corre - sponding t o eac h object , wit h x an d y adjacen t i f an d onl y i f th e correspondin g objects hav e a nonempty intersection .

Interval dimension: Th e interva l dimensio n o f a partia l orde r P i s the minimu m number o f interval order s whic h yiel d P a s their intersection .

Interval filament graph: Th e intersectio n grap h o f a se t o f curve s connectin g end - points o n a fixed line L, where al l curves are required to stay abov e L i n the plane , and betwee n th e tw o endpoints .

Interval graph: Intersection grap h o f intervals o n the line .

Interval number. Th e interva l numbe r o f G i s the smalles t i suc h tha t G ca n b e represented usin g i interval s fo r eac h vertex, wit h vertice s x an d y adjacen t i f an d only i f at leas t on e o f x's interval s has a nonempty intersectio n wit h a t leas t on e o f y's intervals .

Interval order: A partia l orde r i s a n interva l orde r i f element s ca n b e associate d with intervals on the line, such that x < y i f and only i f the interval associated wit h x lie s entirely to the lef t o f the interva l associate d wit h y. 288 Glossary

Inversion-free: A n inversion-free orderin g o f a graph i s an ordering V\V2..v n of the vertices, suc h that fo r al l i < j < k < / , i f there i s an edge fro m vi to Vk and Vj to vi, ther e mus t b e an edg e fro m vi to vi. A graph i s inversion-free i f it admit s a n inversion-free ordering .

Isomorphic: Tw o graphs Gi,G2 ar e isomorphic i f there i s a one-to-one, onto map - ping / fro m vertice s o f G\ t o vertices o f G2, such that x i s adjacent t o y i n G\ i f and onl y i f f{x) i s adjacent t o f(y) i n G2. Grap h isomorphis m i s a famou s ope n problem, in the sense that i t is not known to be either polynomially solvable or NP- complete. A clas s o f graphs i s isomorphism-complete i f solvin g th e isomorphis m problem o n the class with a polynomial tim e algorith m woul d impl y a polynomia l time algorith m fo r general grap h isomorphism .

Isooriented: A set o f objects whos e principa l axe s ar e either mutuall y paralle l o r perpendicular.

Join decomposition: A for m o f graph decompositio n base d o n splits ; a spli t i s a partition o f the verte x se t int o S\ , S% suc h tha t bot h S\ an d $ 2 hav e a t leas t 2 vertices, an d edge s whic h g o fro m S\ t o 5 2 for m a complet e bipartit e graph . I f G ha s a spli t £1 , S2, G i s decomposed int o Si U m, S2 U m, wher e m i s a marke r vertex whic h has edges to exactly thos e vertices whic h have neighbors i n the other partition class . Th e two sets Si U m an d S2 U m ar e then decompose d recursively . k-tree: A graph whic h can be constructed fro m a fc-clique by repeatedly addin g a vertex adjacen t exactl y to some fc-clique in the current tree . k-module: A se t M o f vertice s whic h ca n b e partitione d int o a t mos t k subset s M1..Mk suc h that eac h Mi i s a module i n G - M U M*.

Kf. A clique on i vertices.

Kij: A complete bipartite graph with i vertices in one color class, and j i n the other.

Labeled graph: A labele d grap h ha s name s attache d t o vertices . Fo r example , i f we hav e tw o graphs, on e of which correspond s t o the path 1,2, 3 and the other t o the pat h 1,3,2 , thes e ar e differen t labele d graph s (the y hav e differen t answer s t o the questio n o f whether verte x 1 is adjacen t t o 2) , althoug h th e tw o graphs ar e isomorphic.

Leaf. In an undirected tree , a vertex o f degree 1 . In a rooted tree, a vertex with no children.

Lexicographic breadth first search: An important restrictio n o f breadth firs t search , designed originall y fo r recognizin g chorda l graphs . Initially , al l vertices ar e placed in a single set. Th e algorithm repeatedl y remove s a vertex v fro m th e last curren t set, an d divides al l subsets int o neighbor s o f v and nonneighbors o f v, placin g the subset o f nonneighbors immediatel y befor e the subset o f neighbors. Glossary 289

Line graph: Th e lin e grap h o f G ha s a verte x fo r eac h edg e o f G , an d a n edg e between tw o vertices o f the lin e graph i f and onl y i f the correspondin g edge s hav e a commo n endpoint . H i s a lin e grap h i f an d onl y i f H i s the lin e grap h o f som e graph G.

Line-of-sight graph: Give n a se t o f objects , a line-of-sigh t grap h ha s a verte x fo r each object an d an edge between vertices i f some line segment between these object s avoids all other objects. I n some cases, restrictions are placed on the segments, such as requiring the lin e segment t o be horizontal .

Linear extension: Fo r a partial orde r P , a total orderin g T o f elements o f P suc h that wheneve r x < y i n P, x < y i n T.

Linear matrix: A 0/1 matri x whic h doe s no t contai n an y induce d 2x2 induce d submatrix wit h al l values equal to 1 .

Linear time: Tim e proportional to the siz e o f the input plu s the siz e o f the output .

Linked list: A fundamental dat a structur e i n computer science . I f items ar e store d in a linke d list , reachin g the ith. item i n the lis t require s steppin g throug h eac h o f the i- 1 preceding item s i n the list .

Literal: I n a satisfiability problem , a literal i s either x o r x fo r som e variable x.

Local complementation: Loca l complementatio n o f G a t v change s a grap h b y re - placing the subgrap h induce d b y neighbors o f v with it s complement .

Local information: Informatio n associate d wit h specifi c vertice s o f a graph .

Local Structure: Alternativ e nam e fo r implici t representation . log: logx is defined (i n the tradition o f computer science ) to be the base 2 logarithm of x.

Matching: A se t o f edge s i s a matchin g i f eac h verte x i s a n endpoin t o f a t mos t one edg e i n the set . Computin g a maximu m cardinalit y matchin g ca n b e don e i n polynomial time , bu t finding a n 0(n 2) algorith m fo r matchin g woul d b e consid - ered a n extraordinar y achievement , s o that i f yo u ca n sho w a proble m i s a s har d as maximu m matchin g yo u hav e give n som e evidenc e o f the difficult y o f finding a linear tim e algorithm .

Maximal: A set S i s maximal with respect t o a property i f there i s no set S' whic h properly contain s S an d ha s the property .

Maximal neighbor. A maximal neighbo r o f v i s a neighbo r x o f v suc h that fo r al l neighbors w o f v, N(w) i s a subset o f N[x\. 290 Glossary

Maximum: A set S i s maximum wit h respec t t o a property i f there i s no se t wit h larger cardinalit y whic h ha s the property .

Maximum cut: An NP-complete graph problem, askin g whether there is a partition V\, V2 of the vertice s suc h tha t a t leas t k edge s hav e on e endpoin t i n V\ an d th e other endpoin t i n V^ . Als o known a s max cu t an d simpl e max cut .

Minimal: A set S i s minimal with respect t o a property i f there i s no proper subse t of S whic h ha s the property .

Minimal fill problem: Th e proble m o f finding a set S o f edges to ad d t o G s o that G U S i s chordal, an d there i s no proper subse t S' o f S suc h that GU S f i s chordal. A se t S of edge s i s a minima l fill se t i f an d onl y i f ever y edg e o f S i s the uniqu e chord i n som e 4-eycl e o f G U S.

Minimal separator: Unfortunately , thi s ter m ha s severa l differen t meaning s i n th e field. I n thi s book , a se t S i s a minimal separato r i f there i s some pai r o f vertice s a, b such tha t a and b are i n differen t component s o f G — 5, an d fo r ever y prope r subset S' o f 5 , a an d b are i n th e sam e componen t of G — S. Not e tha t thi s i s not identica l to a cutset S suc h that n o proper subse t o f 5 i s a cutset, whic h i s an alternative usag e o f the term minima l separator .

Minimum: A se t S i s minimum wit h respec t t o a property i f there i s no se t whic h has smalle r cardinalit y wit h th e property . Ever y minimu m se t i s minima l fo r a property, bu t a set ca n be minima l an d no t minimum .

Minimum fill problem: Th e proble m o f minimizing the numbe r o f edges which ca n be adde d t o G s o that th e grap h i s chordal. Th e decisio n versio n i s NP-complete.

Model: Man y o f the grap h classe s discusse d i n this boo k ar e describe d i n terms o f relations (intersection , containment, visibility , and others) betwee n a set o f objects. In suc h cases , w e say that w e are give n a model i f w e are give n the objects , rathe r than bein g give n the grap h i n a standard forma t suc h a s adjacenc y list s o r a n ad - jacency matrix .

Modular decomposition: Alternat e nam e fo r substitutio n decomposition .

Module: A se t M o f vertice s suc h tha t fo r ever y verte x v o f V - M , v i s eithe r adjacent t o ever y vertex o f M, o r v i s adjacent t o n o vertex o f M.

N(v), N[v]: N(u) i s the se t o f vertices adjacen t t o v, an d N[v] includes both v an d N(v).

N-free poset: A partia l order such that th e transitive reduction contains no induced AT, where a n N i s a se t o f fou r vertice s u,v,w,x wit h edge s (w,u), (x,u), (x,v), forming a shape roughl y equa l to the lette r T V i f u and v ar e placed o n top wit h w and x o n bottom . Glossary 291

Neighborhood: Th e se t o f vertices whic h ca n b e reac h fro m a vertex usin g a singl e edge.

NLC-width: Th e NLC-width o f a graph G is the minimum number k o f labels which can b e use d t o construc t G fro m singl e vertex labele d graph s usin g the unio n an d relabel operations . Th e relabe l operatio n assign s al l vertice s wit h curren t labe l i the labe l j. Th e unio n operatio n take s tw o disjoin t graph s G\fi

Node: Alternativ e nam e fo r vertex , use d especiall y i n trees.

Nondetermistically k-decomposable: A graph is nondeterministically /c-decomposabl e with respec t t o a decompositio n i f there exist s a sequenc e o f decompositio n step s which decompose s the graph int o prime component s o f size at mos t k.

NP: A decisio n proble m i s i n N P i f there i s a polynomia l siz e certificat e fo r ever y instance o f the proble m fo r whic h the answe r i s yes.

NP-complete: A proble m X i s NP-complet e i f i t i s i n NP , an d ever y proble m i n NP ca n b e polynomiall y transforme d t o X. Thi s i s a larg e topic whic h canno t b e treated full y i n this book ; reader s unfamilia r wit h the term ar e advise d to consul t [206] fo r mor e information . I n general , NP-completenes s o f a proble m i s use d a s strong evidenc e o f the intractabilit y o f the problem ; a t th e least , a polynomial al - gorithm to solve an NP-complete problem would require an enormous breakthrough. o: f(n ) i s o(g(n) ) i f fo r ever y constan t c there exist s a constan t n o suc h tha t fo r all n > no, f(ri) < cg(n). Intuitively , thi s means that f(n ) grow s more slowl y tha n g(n) eve n i f constants ar e ignored. Fo r the purpose s o f this book , the simple r defi - nition that f(n ) i s o(g(n)) wheneve r f(n ) i s 0(g(n)) bu t g(n ) i s not 0(f(n) ) wil l b e equivalent fo r al l function s studie d here , though th e definition s ar e no t equivalen t in general . A typical us e i s posing the questio n o f finding a n o(f(n) ) algorith m t o solve a problem, wher e the fastes t know n algorith m ha s running tim e B(f(n)) .

0: f(n) i s 0(g(n)) i f there exist constants c , no such that fo r all n > no , f(n) < cg(n). Intuitively, g(n ) grow s at leas t a s quickly a s f(n), i f constant factor s ar e ignored .

Optimization problem: A proble m i n which the goa l i s to find a maximum o r min - imum set . Man y problem s hav e a decisio n (yes/no ) o r optimizatio n versio n o f th e problem.

Order notation: Mathematica l notatio n designe d to compar e growt h rate s o f func - tions, whil e ignorin g constan t factors . I n this book , w e will us e the symbol s o , O , Q, and O , whic h correspon d t o <,<,>,= i f we ignore constant factors .

Ordered chordal graph: Give n a grap h G an d a n orderin g O o f th e vertices , G i s ordered chordal i f there is no chordless cycle C o f length at leas t 4 such that vertice s 292 Glossary occur i n the sam e order i n C a s in O.

Orientation: Assignmen t o f directions to edge s o f an undirecte d graph .

Outdegree: The numbe r o f edges out o f a vertex i n a directed graph .

Outerplanar graph: A graph wit h a planar embeddin g suc h that ever y vertex i s on the oute r face .

Overlap graph: The overla p grap h o f a set o f objects ha s a vertex fo r eac h object , and a n edg e betwee n vertice s i f and onl y i f the object s hav e a nonempty intersec - tion, an d neithe r objec t contain s the other .

Pim. A chordless path o n i vertices .

Parallel module: A module whic h induce s a disconnected graph .

Parent: I n a rooted tree, the next vertex on the path to the root i s called the parent.

Partial k-tree: A (no t necessaril y induced ) subgrap h o f a k-tree.

Partially complemented representation: A form o f representation i n which fo r eac h vertex u , you are given either a list o f neighbors o f v, o r a list o f nonneighbors o f v.

Partial order. Normally , a partia l orde r o n a se t o f element s i s a relatio n whic h is reflexive , transitive , an d antisymmetric . Sinc e thi s boo k deal s graph s withou t self-loops, th e term s partia l orde r an d transitiv e grap h ar e sometime s viewe d a s equivalent.

Partition: Divisio n o f a se t int o subsets , suc h that eac h elemen t i s i n exactl y on e subset. Th e subset s ar e ofte n referre d t o a s blocks o f the partition .

Partitioning: I n general, the process o f subdividing block s of a partition base d on a particular rule . I n this book, w e focus o n partitions which correspond to subsets o f the vertice s o f a graph, an d a subdivision rul e whic h divide s block s int o neighbor s and nonneighbor s o f some vertex x whic h i s not i n the block .

Path: A sequence o f vertices i>i , V2, ••, ^n suc h that eac h Vi is adjacent t o Vi+i.

Path graph: Intersectio n graph s o f undirecte d path s i n undirecte d trees . Differ s from EP T graph s i n the path s whic h shar e a commo n tre e nod e bu t n o commo n tree edge correspond to adjacent vertice s in a path graph, an d nonadjacent vertice s in an EPT graph .

Path orderable graph: A graph whic h admit s a n ordering v\V2.*v n suc h that fo r al l sets o f 3 independent vertice s wit h Vi < Vj < Vk in th e ordering , ever y pat h fro m Vi to Vk goes through a t leas t on e neighbor o f Vj. Glossary 293

Pathwidth: Th e pathwidt h proble m trie s t o minimiz e th e siz e o f th e maximu m clique ove r al l interva l completion s o f a graph . Determinin g whethe r a G ha s a n interval completion with maximum clique size k for input variable k is NP-complete.

Pendant vertex: A vertex o f degree 1 .

Perfect elimination ordering/scheme: Thes e terms , (perfec t eliminatio n schem e and perfec t eliminatio n orderin g ar e use d interchangeably ) refe r t o a n eliminatio n scheme define d b y successiv e remova l o f a simplicia l vertex . A graph i s chordal i f and onl y i f it ha s a perfect eliminatio n scheme .

Perfect elimination bipartite graph: A grap h i n which al l edges can be removed b y repeatedly removin g both endpoint s o f a bisimplicial edge .

Perfect graph: A graph G i n whic h fo r ever y induce d subgrap h H o f G , th e chro - matic numbe r o f H i s equal to the siz e o f the larges t cliqu e i n H.

Perfect matching: A matching i s perfect i f every vertex i s an endpoint o f some edge in the matching .

Perfectly orderable graph: A grap h G whic h ha s a n orderin g O wit h th e propert y that fo r ever y induced subgraph H o f G, greedy coloring o f H whic h color s vertices in the orde r whic h the vertice s appea r i n O produces a n optimal coloring .

Permutation diagram: Intersectio n mode l (a s describe d below ) o f a permutatio n graph.

Permutation graph: Intersectio n grap h o f lin e segments , wher e eac h lin e segmen t connects a point o n lin e L\ wit h a point o n the paralle l lin e L2.

Phylogeny problem: Th e problem o f finding the tree describing the evolution o f dis- tinct specie s fro m a commo n ancesto r species . I n th e perfec t phylogen y problem , distinct specie s ar e assigne d set s o f attributes, an d yo u wan t a n evolutionar y tre e such that eac h interna l nod e i s given a se t o f attributes, an d th e se t o f nodes wit h a give n attribute for m a connected subtre e o f the evolutionar y tree .

PI graph: Intersection graph s o f triangles suc h that eac h triangle ha s one endpoin t on lin e Li, an d tw o endpoints o n the paralle l lin e L2.

PI* graph: Intersection graph o f triangles such that al l endpoints are on two parallel lines Li, L2- Differ s fro m P I graph s i n that som e triangles ca n hav e two endpoint s on Li, whil e others hav e two endpoints o n L 2.

Planar graph: A grap h whic h ca n be drawn i n the plane suc h that n o pair o f edges crosses each other .

Polygon-circle graph: Intersection grap h o f a set o f polygons embedde d i n a circle. 294 Glossary

Poset: Alternat e nam e fo r partia l order .

Postorder. A postorder traversal o f a tree visits all children o f a node before visitin g the nod e itself .

PQ-tree: A tree whic h i s used t o represen t possibl e ordering s o f a set ; element s o f the se t ar e store d a s leave s o f the tree . Th e se t o f lea f descendant s o f a particula r internal node i must occu r consecutively i n the ordering. Interna l nodes are labeled either P o r Q ; childre n o f a P nod e ca n b e permuted i n an y orde r wit h respec t t o the final output list , whil e children o f a Q node are assigned a n ordering, an d mus t appear i n either thi s orde r o r i n reverse orde r i n the output list .

Prime: Indecomposabl e wit h respec t t o a particular for m o f graph decomposition .

Probe interval graph: A grap h i n whic h eac h verte x correspond s t o a n interval , and eac h verte x i s labele d eithe r a s a prob e o r nonprobe . Tw o vertice s ar e adja - cent i f the correspondin g interval s hav e a nonempty intersection , an d a t leas t on e of the vertices is a probe. Thes e arose from a n application i n computational biology .

Projective plane: Althoug h the theory o f projective plane s i s quite a broad subject , they ar e use d i n onl y on e wa y i n thi s book . W e us e th e fac t tha t fo r infinitel y many value s n, i t i s possible to create a 0/1 matri x M wit h n row s and n columns , such that eac h row has ^(n1/2) 1 entries, and eac h pair o f rows ri, r2 has exactly 1 common colum n c such that M[ri , c] — M[r2, c] = 1 .

Proper circular-arc graph: Intersection grap h o f a set o f arcs o n a circle , such tha t no arc contain s anothe r ar c i n the set .

Proper interval graph: Intersection grap h o f a set o f intervals on the line , such tha t no interva l contain s an y othe r interval . Th e clas s i s equivalen t t o uni t interva l graphs.

Pruning sequence: A n elimination schem e defined b y successive removal o f pendant vertices o r one vertex fro m a pair o f twins. G has a pruning sequenc e i f and onl y i f G i s distance hereditary .

Pseudoline: Pseudoline s connec t point s i n the plane via curves, and hav e the prop- erty that eac h pair o f pseudolines cros s each other a t mos t once .

Queue: A data structur e whic h allow s the operation s insertio n an d remova l o f the vertex whic h ha s been i n the queu e fo r th e longes t amoun t o f time.

Quotient graph: Th e quotien t grap h o f a prim e modul e M wit h respec t t o sub - stitution decompositio n ha s on e vertex fo r eac h maxima l prope r submodul e o f M , with tw o vertice s adjacen t i n th e quotien t grap h i f ther e ar e edge s betwee n th e corresponding maxima l submodule s i n the origina l graph . Glossary 295

Random graph: For thi s book , a random grap h i s the resul t o f choosin g eac h edg e to b e i n the grap h o r no t i n the grap h wit h equa l probability .

Realizer. A se t o f linear extension s o f a partial orde r P whic h produc e P a s thei r common intersection .

Recognition problem: Th e problem o f determining whethe r a graph i s a member o f a clas s o f graphs.

Reconstructible vertex: A vertex i s reconstructible fro m a se t o f information i f ad - jacencies t o al l other vertice s ca n b e deduce d fro m thi s information . Thi s i s use d in a more specifi c manne r whe n dealin g with AT-fre e co-AT-fre e graphs , wher e w e specify a se t o f rules whic h allo w the determinatio n o f adjacencies t o a vertex .

Rectangle number: Th e rectangle number o f G is the minimum numbe r k such that every verte x ca n b e associate d wit h k isooriente d rectangles , an d x i s adjacen t t o y i f and onl y i f some rectangle associate d wit h x ha s a nonempty intersectio n wit h some rectangle associat e wit h y.

Rectangle overlap graph: Intersectio n grap h o f isooriente d rectangula r region s i n the plane . W e not e tha t a rectangl e i s considere d a s a regio n o f space , s o that i f one rectangle contains another they are considered to have a nonempty intersection .

Reduced matrix: Th e reduce d M' matri x o f a T-fre e matri x m i s a 0/1 matri x suc h that 1 entries o f M' correspon d to entries o f M whic h are 1 , and ar e not redundan t 1 entries .

Redundant 1: A redundant 1 in a T-fre e matri x i s a 1 entry whic h woul d creat e a T i f the entr y wa s change d t o 0 . Thes e I s ar e calle d redundant , sinc e i t i s unnec- essary to store the positions o f redundant I s in our representation o f T-free matrices .

Refinement: Th e process o f taking a partition o f a set, an d gettin g a new partitio n by dividing set s into subsets .

Representation problem: Th e problem o f finding a form o f representation fo r a class of graphs whic h satisfie s som e conditio n (e. g finding a n implici t representation , o r finding a space-optima l representatio n whic h allow s constan t tim e adjacenc y test - ing). Thi s i s separate fro m th e construction problem , i n that i t deal s only with th e existence o f suc h a representatio n rathe r tha n constructio n o f representation s fo r individual graphs .

Representative graph: Alternate nam e fo r quotien t graph .

Robust algorithm: A robust algorith m fo r solvin g a problem P o n an input clas s C is required to always answer P correctl y when the input i s in class C, and when the input i s not in C may either answer P correctl y or answer that the input i s not in C.

Rooted tree: A rooted tre e i s a directed graph , whic h has a n underlying tre e an d a special vertex r calle d the root. Direction s ar e assigne d s o that ever y vertex o f the 296 Glossary tree i s reachable vi a a directed pat h fro m r .

Sandwich problem: Give n a se t o f edge s E\ an d a se t o f edge s E2 th e sandwic h problem fo r a clas s o f graphs ask s whethe r ther e i s any grap h G i n the clas s suc h that G contains ever y edg e fro m E\, an d ever y edg e i n G comes fro m E2.

Satisfiability: Th e proble m o f determinin g whethe r a Boolea n formul a give n i n conjunctive norma l for m ha s a trut h assignmen t suc h tha t th e entir e formul a i s evaluated a s true. Thi s wa s the first proble m show n to b e NP-complete .

Semiorder. A partial orde r suc h that eac h elemen t e can b e mappe d t o a numbe r w(e), and there i s a fixed threshold t such that x < y i f and onl y i f w(y) — w(x) > t.

Separator. Alternat e nam e fo r cutset .

Series module: A module whic h induce s a subgraph M suc h that th e complemen t of M i s disconnected .

Series-parallel graph: Although there ar e slightly differen t definition s co-existin g i n the literature , i n this boo k a graph i s series-parallel i f all edges can b e removed b y a sequenc e o f removal o f degree 1 vertices, and remova l o f degree 2 vertices v wit h neighbors w,x followe d b y additio n o f an edg e w,x i f such a n edg e doe s no t exis t already.

Series-parallel poset: A partial orde r whic h ca n be constructed fro m singl e elemen t partial order s usin g the operations disjoin t unio n an d join, wher e join make s ever y element o f Pi < ever y elemen t o f P2. Equivalen t t o partial order s with underlyin g cographs.

Sibling: Vertice s i n a rooted tre e whic h hav e a common parent .

Sign pattern: A sig n patter n fo r a sequenc e pi,,..,^ o f polynomia l function s i s a sequenc e si , ..,Sk o f I s an d -I s suc h tha t ther e i s som e assignmen t t o variable s which make s ever y pi positiv e i f and onl y i f s ^ = 1 .

Simple matrix: A matrix suc h that n o pair o f columns i s identical .

Simple vertex: A vertex v suc h tha t neighbor s o f v induc e a clique , an d fo r eac h pair o f neighbors w,x o f v eithe r N(i> ) C N[iu ] o r N(u> ) C N[f] .

Simplicial vertex: A vertex v suc h that N[v ] induces a clique.

Sink: A vertex i n a directed grap h o r partial orde r whic h ha s n o outedges.

Skew partition: A skew partition o f a graph i s a partition o f the vertex set into fou r nonempty subse t A,B,C,D suc h tha t ther e i s a n edg e fro m ever y verte x o f A t o every vertex o f B, an d there is no edge from an y vertex of C to any vertex o f D. Re - cursive decompositio n usin g ske w partition finds a ske w partition A,B,C,D, an d recursively decompose s the subgraphs induce d b y A , B, C, b y A , £?, D1 b y A, C, D, Glossary 297 and b y B, G , D.

Source: A verte x i n a directe d grap h o r partia l orde r whic h ha s n o inedges .

Space optimal: A representation o f a graph clas s which has f(n) graph s o n n vertice s is space optima l i f i t use s 0(log(f(n) ) bit s t o stor e eac h grap h i n the class .

Sparse graph: A grap h wit h a smalle r numbe r o f edges . Th e ter m i s informal ; i n some case s sparse may mean o(n 2), i n other case s i t coul d be a s restrictive a s 0(n).

Sphericity: Th e sphericit y o f G i s the smalles t dimensio n suc h tha t G ca n b e rep - resented a s th e intersectio n grap h o f d-dimensiona l spheres . Ever y grap h i s th e intersection grap h o f sphere s i n som e dimensio n [360] ; thi s contrast s wit h th e fac t that ther e ar e partia l order s whic h ar e no t containmen t graph s o f sphere s i n an y dimension. [187 ]

Spider graph: Alternat e nam e fo r polygon-circl e graph .

Split: A partition o f vertices o f a graph int o Si , 5* 2 is a split i f it meets the followin g conditions. 1 ) Si an d S2 hav e a t leas t tw o vertices . 2 ) Le t Su n o f vertice s i n Si with a t leas t on e neighbo r i n S2, an d le t S2i n b e th e se t o f vertice s i n S 2 wit h a t least on e neighbo r i n Si. Ever y verte x o f Su n mus t b e adjacen t t o ever y verte x o f S2in- Not e that thi s i s a specia l cas e o f the origina l definitio n o f spli t i n [138 , 139] , which wa s define d fo r directe d graph s a s wel l a s undirecte d graphs .

Split decomposition: Alternat e nam e fo r joi n decomposition .

Split graph: G i s a spli t grap h i f th e vertice s o f G ca n b e partitione d int o S, K, where S induce s a n independen t se t i n G an d K induce s a cliqu e i n G ; i.e. , G ca n be obtained fro m a bipartite grap h b y making on e colo r clas s a clique . Spli t graph s are exactl y thos e graph s whic h ar e bot h chorda l an d co-chordal .

Stable set: Alternat e nam e fo r independen t set .

Stack: A data structur e whic h allow s th e operation s insert , an d remov e th e objec t which ha s bee n i n the se t fo r th e shortes t perio d o f time .

Star. Ki^ fo r an y valu e o f i > 1 .

Steiner tree: Th e Steine r tre e proble m take s a grap h G an d a subse t S o f vertice s as input , an d ask s fo r th e minimu m subtre e T o f G whic h connect s al l vertice s i n S. Th e decisio n versio n o f th e Steine r tre e proble m i s NP-complete , eve n i n th e unweighted case .

Star cutset: A sta r cutse t o f a grap h i s a se t C suc h tha t G - C i s disconnected , and som e verte x c in G i s adjacen t t o ever y othe r verte x o f G .

String graph: Intersectio n grap h o f curve s i n the plane . 298 Glossary

Strong perfect graph conjecture: Famou s problem i n graph theory , stating that G is perfect i f and onl y i f G i s a Berg e graph . A proof o f the conjectur e ha s just bee n announced. [102 ]

Strongly chordal graph: Although properly speaking this is a characterization rathe r than th e origina l definitio n o f the class , w e will say a graph i s strongly chorda l i f it has a n eliminatio n schem e define d b y successiv e remova l o f simple vertices .

Strongly connected graph: A directe d grap h i n whic h ther e i s a pat h fro m ever y vertex to ever y othe r vertex .

Substitution decomposition: A for m o f grap h decompositio n whic h recursivel y de - composes a graph int o connected component s i f G i s disconnected, connecte d com - ponents o f G i f G i s disconnected, an d maxima l prope r submodule s o f G i f both G and G are connected .

Threshold graph: A grap h i n whic h eac h verte x v va n b e assigne d a weigh t w(v) and tw o vertices x , y ar e adjacen t i f w(x) + w(y) > t fo r som e fixed threshold t.

Tolerance graph: A graph i n whic h ever y verte x ca n b e assigne d bot h a n interva l and a tolerance, such that x an d y are adjacent i f and onl y i f the intersection o f the intervals exceeds the minimum o f the two tolerances. Interva l graphs correspond t o tolerance graphs with tolerance 0. N o implicit representation i s known for this class.

Topological sort: A n orderin g o f vertices o f a directe d acycli c graph, suc h that fo r every directed edg e (x,y) , x precede s y i n the ordering .

Total interval number. Th e tota l interva l numbe r o f a grap h G i s th e smalles t number o f intervals suc h that eac h interval i s associated wit h exactly on e vertex ( a single verte x ca n b e associate d wit h multipl e intervals) , an d x i s adjacen t t o y i f and onl y i f som e interva l associate d wit h x ha s a n intersectio n wit h som e interva l associated wit h y.

Totally balanced: A hypergraph i s totally balance d i f every cycl e o f length greate r than 2 has a n edg e containin g a t leas t 3 vertices o f the cycle . A well-known theo - rem i s that a hypergraph i s totally balance d i f and onl y i f every subhypergrap h i s a hypertree .

Transitive closure: Give n a directed grap h G , the transitive closur e o f G is a graph on th e sam e verte x set , wit h a n edg e fro m x t o y i n th e transitiv e closur e i f an d only i f there i s a path fro m x t o y i n G .

Transitive graph: A directed graph G such that wheneve r there are edges (x , y) an d (y, z\ (x , z) i s also an edge .

Transitive orientation: A n assignmen t o f direction s t o edge s o f a comparabilit y graph, suc h that th e resultin g directio n o f edges i s transitive. Glossary 299

Transitive reduction: Give n a directed acycli c graph G, the transitiv e reductio n o f G i s a graph o n the sam e verte x set , wit h a n edg e fro m x t o y i f x ha s a n edg e t o t/inG, an d ther e i s no path o f length greate r tha n 1 from x t o y i n G.

Trapezoid graph: Intersectio n grap h o f trapezoids , wher e eac h trapezoi d ha s tw o endpoints o n a lin e L\ an d tw o endpoints o n a parallel lin e L2 . Not e that L\ an d L2 are the sam e fo r al l trapezoidal objects .

Tree: A connected acycli c graph .

Treewidth: Th e origina l definitio n o f treewidth deal s with a for m o f tree decompo - sition. Fo r this book, i t i s easier to understand a characterization; G has treewidt h k i f and onl y i f G i s a partial k-tree. Th e fac t tha t mino r close d classe s whic h d o not contai n al l planar graphs must hav e bounded treewidth ha s deep consequences; see [412] . Determinin g th e treewidt h o f a grap h i s NP-complete , althoug h deter - mining whethe r a graph ha s treewidth k i s polynomial fo r fixed k. Th e treewidt h problem can also be viewed as minimizing the maximum cliqu e size over all chordal completions o f a graph .

Triangle-extendible: A graph i s triangle-extendible i f the vertices can be ordered s o that fo r ever y triangle u < v < w, ever y vertex which comes after w and i s adjacen t to both v an d w i s also adjacent t o u; in other words , fo r ever y K4 - a single edge , the nonadjacen t vertice s ar e not bot h firs t an d las t i n the ordering .

Twins: A pair o f vertices x,y suc h that N(x ) = N(y ) i f x an d y ar e nonadjacent , and N[x ] = N[y ] i f x an d y ar e adjacent .

Two-pair: A pair o f vertices x, y such that ever y chordless path between x and y has length 2 . Equivalentl y fo r a connected graph , removin g N(x) D N(y) disconnect s x and y.

Underlying graph: The underlyin g grap h o f a directed grap h o r pose t i s the grap h formed b y removing direction s fro m edges .

Uniform cost assumption: Th e assumptio n tha t arithmeti c operation s involvin g numbers o f polynomia l siz e take s constan t time . Thi s i s a standar d simplifyin g assumption mad e i n analysi s o f algorithms .

Unit circular-arc graph: Intersectio n graph s o f arc s wit h identica l length s i n th e circle. Unit disk graph: Intersection grap h o f disks with equa l diameters i n the plane .

Unit interval graph: Intersectio n graph s o f interval s o f identica l siz e i n th e line . These ar e equivalen t t o proper interva l graphs .

Universal graph: A graph G i s (vertex-induced ) universa l fo r a class C o f graphs i f every graph i n C i s an induced subgraph o f G. Th e concep t o f a universal graph i n the non-induced sens e also exists in the literature, but doe s not appl y to this book . 300 Glossary

Universal vertex: A vertex whic h i s adjacent t o ever y other verte x i n the graph .

Vertex cover. A se t S o f vertice s suc h tha t ever y edg e o f th e grap h ha s a t leas t one endpoin t i n S. Decidin g whethe r G ha s a verte x cove r o f inpu t siz e k i s NP- complete. I n the optimization version , the problem ask s fo r the siz e o f the smalles t vertex cover .

Vertex expansion: Th e operatio n o f replacing vertex v b y a clique C, eac h membe r of which ha s the sam e adjacenc y t o other vertice s o f the grap h a s v.

Vertex multiplication: Th e operatio n o f replacin g verte x v b y a n independen t se t J, eac h member o f which has the same adjacency t o other vertices of the graph as v.

Visibility graph: A graph i n which vertices can be mapped t o corners o f a polygon , such that vertice s ar e adjacen t i f they ar e eithe r neighbor s o n the polygon , o r th e line segment connectin g the tw o vertices i s entirely insid e the polygon . von Emde Boas trees: A data structure whic h allow s insertion, deletion , an d searc h for predecesso r i n the curren t se t ove r a se t wit h element s i n the rang e l..n . Th e trees use O(loglogn) tim e per operation rather than the O(logn) time per operatio n of such structures a s 2-3-tree s designe d t o handl e thes e operations o n sets with n o upper boun d o n range .

Warren's theorem: Thi s important theorem , discussed i n chapter 4 , was designed to give an upper bound o n the number o f sign patterns o f a set o f polynomials. I n this book, it is used to give an upper bound on number o f graphs in a class of graphs. Fo r various classes o f graphs, vertices can be made to correspond wit h sets o f variables, and adjacenc y betwee n vertice s x an d y i s determined b y the sig n o f a polynomia l equation on the variables corresponding to x, y, and possibly other vertex variables.

Weak order: A partia l orde r i n whic h ever y elemen t v ca n b e assigne d a numbe r w(w), an d x < y i f w(x) < w(y). I n othe r words , thes e ar e order s whic h ca n b e derived fro m tota l order s b y substituting independen t set s fo r elements .

Weakly chordal graph: A grap h G with n o induce d cycle s o f length greate r tha n 4 in either G o r G.

Weakly triangulated graph: Alternate nam e fo r weakl y chorda l graph .

Well-covered graph: A grap h i n which al l maximal independen t set s have the sam e cardinality.

Width: Th e widt h o f a partial orde r i s the maximu m independen t se t (als o know n as antichain i n this context ) i n the underlyin g graph .

0/1 matrix multiplication: Th e resul t o f performin g matri x multiplicatio n A x B for matrice s A an d B wit h al l entrie s equa l t o 0 o r 1 , an d replacin g al l nonzer o entries o f the resul t b y Is . Glossary 301 a: I f A an d B ar e n b y n matrices , th e tim e o f the fastes t matri x multiplicatio n algorithm fo r computin g Ax B know n i s written a s 0(n a). Whe n thi s boo k wa s written, a wa s approximately 2.376 .

T: A pai r o f row s r\ < T2 and column s c\ < C2 i n a 0/ 1 matrix , suc h tha t th e submatrix induce d b y these column s ha s I s i n every entry excep t positio n r2,C2 -

Q: Intuitively, f(n ) i s fi(g(n)) mean s that f(n ) grow s at leas t a s quickl y a s g(n), i f constants ar e ignored. Ther e ar e several differin g precis e definitions , whic h ar e no t equivalent i n general but ar e equivalent fo r al l functions i n this book. M y preferre d definition i s that f(n ) i s fi(g(n)) i f there i s a constan t c such that f(n ) > g(n ) fo r infinitely man y value s o f n , bu t fo r thi s boo k i t i s sufficien t t o sa y tha t f(n ) i s Q(g(n)) wheneve r g(n ) i s 0(f(n)) .

©: f(n ) i s G(g(n) ) i f f(n ) i s 0(g(n) ) an d g(n ) i s 0(f(n)) . Thi s i s ou r notio n o f equality i n order notation , i.e . i f constants ar e ignored .

2-join: Fundamentally , a partitio n o f the verte x se t suc h tha t edge s betwee n th e set ca n b e divide d int o tw o complet e bipartit e graphs . Ther e ar e severa l variants , depending on whether these complete bipartite graphs are allowed to share vertices, and on conditions fo r ruling out trivial partitions. Thus , there are definitions whic h require only that eac h set contains at leas t 3 vertices, while others d o not allo w one side o f the partitio n t o b e a n induce d path . Som e definition s als o requir e a pat h within each side o f the partition connecting the different complet e bipartite graphs . This page intentionally left blank Survey o f Results o n Grap h Classe s

This chapte r wa s suggeste d b y on e o f th e referees , wh o note d tha t man y al - gorithms fo r dealin g wit h specifi c classe s o f graphs ar e discusse d i n the book , bu t that thes e are difficul t fo r a reader intereste d i n a single class o f graphs to discover . The numbe r o f graph classe s an d result s o n thes e classe s ar e trul y overwhelming , so rather tha n tr y t o organiz e thes e i n the for m o f a tabl e wit h grap h classe s o n one side and al l possible optimization problem s o n the other, I have chosen to giv e a ver y brie f sketc h o f what I believe to b e som e o f the mos t importan t result s an d open problem s i n algorithms an d representatio n fo r a number o f classes o f graphs. An algorithmi c surve y o n variations o f the dominatin g se t proble m i s given i n [329]; I will restrict m y treatment her e to the basi c dominatin g se t proble m unles s there i s a particular reaso n to cove r othe r dominatio n problems . Le t m e note tha t proving NP-completenes s o f minimum fil l wa s a longstanding ope n proble m befor e being show n i n [491] . Ther e i s fa r les s wor k showin g tha t minimu m fil l i s NP - complete fo r a clas s tha n i s th e cas e fo r othe r problems , give n th e difficult y o f showing NP-completenes s a t all ; i f ther e i s n o commen t o n th e fil l problem , thi s generally mean s that ther e i s n o polynomia l tim e algorith m t o fin d th e minimu m fill o n thi s clas s o f graphs , an d th e clas s doe s no t see m t o b e nea r th e borde r o f what w e know to b e poly normally solvable . There i s a n extensiv e bod y o f literatur e dealin g wit h paralle l algorithm s o n these grap h classes ; sinc e my are a o f research i s sequential algorithms , I wil l leav e a surve y o n paralle l algorithm s t o som e othe r author . A s oppose d t o algorithm s discussed i n the rest o f the text, I did not try to verify that al l the algorithms liste d here ar e correct . There ar e man y importan t resource s availabl e fo r findin g algorithm s o n grap h classes. Th e boo k [74 ] has a tabl e dealin g wit h bes t tim e complexitie s fo r recog - nizing a muc h large r se t o f graph classe s than i s discussed here ; corrections to thi s table ar e currently kep t a t www.informatik.uni-rostock.de/~ab/survey/errata.html. At the sam e site, a very nic e program fo r question s o f inclusions between classe s o f graphs ca n b e foun d a t www.informatik.uni-rostock.de/~gdb/isgci/isgci.html. A table o f graph clas s inclusions an d counterexample s specializin g i n subclasses o f perfect graph s i s available on-lin e a t www.informatik.hu-berlin.de/~hougardy/paper/classes.html. A surve y dealin g wit h polynomia l algorithm s fo r som e fundamenta l NP-complet e optimization problem s o n variou s classe s o f graph s appeare d i n [282] , an d som e updates appeare d i n [283] . A n ambitiou s attemp t t o maintain a n on-lin e referenc e

303 304 Survey o f Results o n Graph Classe s list o f best know n algorithm s fo r problem s o n graph classe s wa s attempted an d some partial result s can be seen at http://web.cs.ualberta.ca/~Stewart/GRAPH/index.html. This projec t becam e muc h to o large fo r a singl e perso n t o maintain; th e current plan i s to distribute th e work to individuals wit h goo d knowledg e o f a particula r class o f graphs. I f you are interested i n helping wit h thi s project , i t woul d b e a great hel p to all of us!

AT-FREE GRAPHS : There are 2e(n ) graphs without asteroida l triples, so represen- tation i s not an issue. A straightforward algorith m ca n recognize AT-fre e graph s in 0(n 3) time. Usin g fas t matri x multiplicatio n a s a subroutine, thi s ca n be re- duced t o 0(n 2,79) tim e [332] . I t wil l b e difficul t t o reduc e th e cost t o les s tha n the cos t o f a singl e matri x multiplication , sinc e th e proble m i s a s har d a s rec - ognizing triangle-fre e graphs , a s discussed i n the text. Polynomia l algorithm s on asteroidal triple-fre e graph s includ e a n 0(n3) algorithm fo r weighted independen t set [316] , base d o n the algorithm o f [81] , an 0(n3) Steine r tre e algorith m [26], and a n 0(n6) unweighte d dominatin g se t algorith m [330] ; weighte d dominatin g set i s NP-complete o n the subclass o f co-comparability graphs . Cliqu e and clique cover ar e NP-complete o n this clas s [81] ; these problem s separat e AT-fre e graph s in complexit y fro m co-comparabilit y graphs . Treewidt h an d minimum fill are also NP-complete o n AT-free graphs , as a consequence o f the original NP-completenes s reduction for these problems [24 , 491], but are polynomially solvable for the subclass of AT-fre e co-AT-fre e graph s [316] . Bandwidt h i s NP-complete fo r the subclass of co-bipartite graphs ; a n approximatio n algorith m fo r AT-fre e graph s graph s wit h performance rati o 2 is given in [308] . Ma x cut is also NP-complete fo r co-bipartite graphs [47] , and co-bipartite graphs are isomorphism-complete. Chromati c numbe r and Hamilton cycl e are open for AT-free graph s at this time. Feedbac k vertex set, which is polynomial fo r co-comparability graphs , doe s not seem to have been stud - ied for AT-free graphs. A very simple linear time algorithm for finding a dominating pair in an AT-free grap h is given in [125] ; finding a robust linea r time algorithm for this proble m i s open. Findin g a simpl e intersectio n representatio n whic h exactl y characterizes the class is an open problem .

BOUNDED TOLERANC E GRAPHS : Th e proble m o f recognizin g bounde d toleranc e graphs i n polynomial tim e i s open. A n implicit representatio n ca n be constructed in polynomia l time , thank s t o the fac t tha t ever y bounde d toleranc e grap h i s a trapezoid graph . Althoug h mos t algorithm s o n these graph s ar e based upo n thei r containment i n larger graph classes , several papers hav e been written solvin g prob- lems on bounded toleranc e graphs i f the model is given as input. Ther e are papers for solvin g the dominating se t and Hamilton cycl e problems on bounded toleranc e graphs mor e efficientl y (0(n 2) fo r dominating se t and O(nlogn) fo r Hamilton cy- cle) tha n o n the larger clas s o f co-comparability graph s i f a model i s given [3 , 4] , and a paper givin g an 0(n3log2n) algorith m fo r dominating set in the complement of bounde d toleranc e graph s [296] . Th e last pape r mentione d i s an example o f a problem whic h can be solved efficientl y i f a model i s given a s input, but is open if the input i s given in adjacency matri x form . I know o f no NP-completeness result s specifically fo r bounded toleranc e graphs ; thus, we know onl y result s implie d fro m Survey o f Results o n Graph Classe s 305 containment o f suc h classe s a s interval graph s an d permutation graphs . Isomor - phism i s open fo r bounded toleranc e graphs , bot h wit h the model give n and when it i s not. BOXICITY: Recognitio n o f boxicit y k graph s fo r k > 1 is NP-complete . Boxic - ity k graph s hav e a n implicit representatio n b y definition, bu t finding a n implici t representation i n polynomia l tim e i s open . Cliqu e ca n be solve d i n polynomia l time (O(nlogn ) tim e give n the model fo r boxicity 2 [276] ) on this clas s eve n i f the model i s not given, sinc e the number o f maximal clique s i s polynomially bounded . Clique cove r [200] , 3-colorin g [337 ] and independent se t [408 ] have bee n show n to be NP-complete on boxicity 2 graphs. Containmen t o f grid graphs and trees show s that Hamilto n cycle , dominating set, Steiner tree, and bandwidth ar e NP-complete for boxicit y 2 graphs. A n O(logn) approximatio n algorith m fo r independent se t on boxicity graphs is given in [6]; [174] notes that this achieves the same approximatio n ratio fo r weighted independen t set . Feedbac k verte x set is NP-complete o n graphs with bounde d boxicity , sinc e ever y plana r grap h ha s boxicity a t mos t 3 [458] ; thi s may be open fo r boxicity 2 graphs. A s far as I could determine , the complexity of such problem s a s max cut, isomorphism, an d treewidth i s unknown fo r graphs o f bounded boxicity . However , i t i s possible tha t som e classe s fo r whic h thes e prob - lems are known to be difficult hav e bounde d boxicity ; fo r example, I am not sur e whether suc h classe s as degree 3 graphs an d line graphs ca n have arbitraril y larg e boxicity. CHORDAL GRAPHS : Chorda l graph s ca n be recognized i n linear time , a s discussed in the text. A n intersection representatio n a s subtrees o f a tree ca n also be foun d in linea r time . Man y problem s ar e solvabl e efficientl y i n polynomia l tim e fro m the perfec t eliminatio n schem e o r cliqu e tre e model . Weighte d clique , weighte d independent set , coloring, an d clique cove r ca n all be solved i n linear tim e i n this fashion. Man y problem s whic h ar e NP-complete o n chordal graphs , suc h as domi- nating set, Steiner tree, bandwidth [306 ] max cut [47 ] and Hamilton cycl e [115 ] ar e shown to be NP-complete o n the smaller clas s o f split graphs ; grap h isomorphis m is also isomorphism-complete o n both classe s [54] . Treewidth an d minimum fill are trivial o n chordal graphs . Treewidt h ca n be solved i n polynomial tim e i f there are no chordles s cycle s o f length large r tha n k fo r any fixed k [52] ; maximum weigh t induced path s ca n also be found i n polynomial tim e o n this superclas s o f chordal graphs [214] . Th e weighted feedbac k verte x problem i s solvable in polynomial tim e for chorda l graphs ; thi s come s fro m th e fact tha t feedbac k verte x se t on a chorda l graph correspond s to finding a minimum se t of vertices containing at least on e ver- tex fro m eac h triangle , an d this i s a special cas e o f a coverin g proble m solve d fo r chordal graph s i n [121] . CHORDAL BIPARTIT E GRAPHS : A S discussed i n th e text , chorda l bipartit e graph s can b e recognized i n 0(min{n 2, mlogn}) time . Existenc e an d construction o f an implicit representatio n ar e open fo r this class . NP-complet e problem s o n chorda l bipartite graphs include dominating set, Hamilton cycl e (these two problems can be solved in polynomial time on the subclass o f convex graphs [150 , 378]) Steiner tre e [379, 378] , and bandwidth; chorda l bipartite graphs are also isomorphism-complet e [380]. Th e feedback verte x problem , whic h i s polynomial fo r conve x graph s [347] , does not seem to have been studied fo r chordal bipartite graphs . Treewidt h can be solved in 0(ma) time , where a i s the exponent fo r matrix multiplicatio n [305] ; the related proble m o f pathwidth (correspondin g t o minimizin g th e maximum cliqu e 306 Survey o f Result s o n Grap h Classe s size in an interval completion, rather than a chordal completion, o f the input graph ) is NP-complete fo r chorda l bipartit e graph s [303] . Minimu m fill ca n b e solve d i n 0(n4) tim e [94] . Man y problems , suc h a s matching, computin g neighborhoo d con - tainments, independent set , shortest paths can be solved more simply and efficientl y on chordal bipartite graphs than on general bipartite graphs using the T-free matri x characterization, a s discusse d i n the text . CIRCLE GRAPHS : Circl e graph s ca n b e recognize d i n 0(n2) tim e [442] . Thi s algo - rithm als o constructs th e model , whic h i s an implici t representation . Sinc e neigh - bors o f a vertex for m a permutation graph , cliqu e i s clearly solvabl e i n polynomia l time; the best know n complexity i s O(nlogn+min{ra,no;}) fo r the unweighted cas e [364], an d 0(nlogn+min{n 2,mloglogn}) fo r th e weighte d cas e [20] , assumin g th e model is given as part o f the input. I t i s also not difficul t t o devise polynomial tim e independent se t algorithm s fo r thi s class , with 0(n 2) th e bes t know n tim e bound . For the unweighte d cas e this i s attributed t o Buckingha m i n a 198 1 manuscript a t Stewart's website listed i n the introduction, bu t I could not find a published versio n achieving the sam e time bound earlie r than [20] , which achieve s this bound fo r th e weighted case as well. Dominatin g set i s NP-complete fo r this class [295] ; a 2-he ap- proximation algorith m i s given in [277] . Th e /c-colorin g problem i s NP-complete fo r k > 3 [208 , 471]; an O(nlogn) algorith m fo r the 3-colorin g proble m on circle graph s is give n i n [472] . Not e tha t ther e i s som e disagreemen t i n th e literatur e o n rela - tionship between cliqu e size and chromatic number i n circle graphs; the paper [471 ] claims that chromati c numbe r i s at mos t twic e the cliqu e size fo r circl e graphs, bu t more recen t paper s b y experts i n the field giv e 2 U a s the bes t know n uppe r boun d on chromati c numbe r i n terms o f clique siz e [319] . Th e Hamilto n cycl e problem i s NP-complete on circle graphs [147] ; note that thi s is given as solvable in polynomial time in [282] , but this is a case of a misplaced column entry in the table. Bandwidt h is NP-complete, sinc e every tree i s a circle graph. Schaffe r gav e Johnson a n outlin e of a Steiner tree algorithm fo r circl e and circular-ar c graphs (an d thus the proble m is listed a s polynomiall y solvabl e i n [282]) ; [40 ] will be the first generall y availabl e polynomial algorithm fo r Steine r tree on circle graphs. Othe r polynomial time solv- able problem s o n circl e graphs includ e minimu m fill and treewidth , whic h ca n b e solved i n 0(n 3) tim e [302 , 311] , and isomorphism , whic h ca n b e solve d i n O(nra ) time [265] . I wa s unabl e t o find an y wor k o n cliqu e cove r fo r circl e graphs ; sinc e both cliqu e cove r an d circl e graphs ar e heavil y studied , thi s ma y b e a n interestin g open problem . CIRCULAR-ARC GRAPHS : Thi s i s a well-studie d class , an d man y mor e algorithm s have been designed specifically fo r circular-arc graphs than I will list here. Circular - arc graph s ca n b e recognize d b y a n algorith m whic h als o construct s th e mode l i n linear tim e [365] . Maximu m cardinalit y independen t se t ca n b e solve d i n O(n ) time, i f th e inpu t i s give n a s a circularl y ordere d lis t o f endpoint s [225 , 363] , a s can minimu m cardinalit y dominatin g se t [271] . Th e bes t know n tim e boun d fo r the weighte d dominatin g se t proble m i s 0(n+ra) [92] . I kno w o f n o pape r whic h specifically solve s th e weighte d independen t se t proble m fo r circular-ar c graph s ([188] solves the problem in 0(n2) tim e for the superclass of circle trapezoid graphs), though it is obviously solvable from the model in 0(ln) tim e where I is the minimu m load o n the circle , i.e . th e numbe r o f arc s passin g throug h th e poin t o n the circl e which i s covered by the fewes t arcs . Maximu m cliqu e i s solvable given the model i n O(n) tim e [14] . Weighte d cliqu e ca n b e solve d i n 0(nlogn+raloglogn ) tim e [433] . Survey o f Result s o n Grap h Classe s 307

Hamilton cycl e can b e solve d i n 0(n 2logn) tim e o n circular-ar c graph s [432 ] (not e that th e linea r an d 0(n 2) tim e Hamilto n circui t algorithm s cite d a s forthcomin g in [346 ] wer e flawed ) an d cliqu e cove r i n O(rc ) tim e [271] . Isomorphis m ca n b e solved i n O(nra) tim e [265] ; at on e point, I was part o f a paper claimin g a n 0(n 2) bound fo r this , bu t ou r algorith m relie d o n a resul t fro m anothe r pape r whic h may b e i n error . Colorin g i s NP-complet e o n circular-ar c graphs , a s discusse d in th e text . I t i s possibl e t o find a 3/ 2 optima l colorin g [292] , an d t o optimall y solve th e /c-colorin g proble m fo r constan t k i n O(n ) tim e [208] ; ther e i s als o a randomized polynomia l algorith m wit h a n approximatio n rati o o f 1+1/ e i f th e number o f color s i s not O(logn ) [335] . Bandwidt h i s NP-complete fo r circular-ar c graphs; an approximation algorith m with (optimal ) performanc e rati o 2 is given i n [334]. Othe r problem s solved fo r circular-arc graphs include an O(nlogn) algorith m for matchin g whe n th e mode l i s give n [348] , an d 0(n 3) algorithm s fo r minimu m fill [311 ] and treewidt h [451] . Althoug h I have generally limite d mysel f to coverag e of sequentia l algorithms , not e tha t unweighte d independen t set , cliqu e cover , an d dominating se t ca n b e solve d optimall y give n the mode l i n parallel, usin g O(logn ) time o n 0(n/logn ) processor s [404] . Maximu m cu t seem s to b e ope n fo r circular - arc graphs , a s wel l a s th e subclas s o f interva l graphs . Th e statu s o f Steine r tre e is slightly unclear ; Schaffe r sketche d a proof that thi s i s polynomially solvabl e (fo r both circle and circular-arc graphs), and thus it appeared a s polynomial in the table of [282] , though n o algorithm solvin g the problem appears i n the general literature . I foun d n o papers dealin g wit h feedbac k verte x se t i n circular-ar c graphs , bu t th e problem ca n b e solve d i n polynomia l tim e b y reductio n t o feedbac k verte x se t o n interval graphs. Pathwidt h i s open fo r circular-ar c graphs . CLIQUE-WIDTH K GRAPHS: Th e recognitio n proble m ca n b e solve d i n polynomia l time fo r clique-widt h 1 (trivial), 2 (equal s cograp h recognition) , an d 3 [122] , an d is ope n i n genera l an d fo r othe r fixed value s of k. A polynomia l tim e algorith m which construct s a n O(logn ) clique-widt h decompositio n tre e fo r an y grap h wit h clique-width a t mos t k fo r fixed k i s give n i n [281] . Fo r an y fixed /c , an implici t representation ca n b e foun d i n polynomia l tim e usin g balance d £>homogeneou s sets, a s discusse d i n th e text . Th e ke y resul t o n thi s clas s i s tha t an y problem s posed i n monadi c secon d orde r logi c wit h quantificatio n ove r verte x sets , bu t n o quantification ove r edg e sets, can be solve d i n linear tim e o n clique-width k graph s if th e decompositio n tre e i s give n a s par t o f th e inpu t [135] . Problem s solvabl e in thi s wa y includ e independen t set , dominatin g set , z-colorabilit y fo r fixed z , an d various other NP-complet e an d polynomiall y solvabl e problems fo r genera l graphs . Certain problems which are not expressible in this form o f logic can also be solved on clique-width k graphs; such problems include Hamilton cycle and max cut [478] , and chromatic numbe r whe n the number o f colors may be arbitrarily larg e [313] . I f the graph i s given in adjacency matri x form, independen t se t and i-colorability ca n still be solve d i n polynomial tim e [57] , but th e questio n o f whether al l problems pose d in th e restricte d secon d orde r monadi c logi c b y robus t algorithm s remain s open . Most othe r problem s hav e no t bee n investigate d fo r graph s wit h bounde d clique - width. Thus , fo r example , th e complexit y o f isomorphis m o n graph s o f bounde d clique-width woul d see m to be an open problem, both wit h the decompositio n tre e given an d fo r robus t algorithms . COGRAPHS: Cograph s ar e a particularl y tractabl e clas s o f graphs . The y ca n b e recognized i n linear time , and a cotree can be constructed i n the sam e time boun d 308 Survey o f Results o n Graph Classe s

[128]. Sinc e this cotree is unique, cograph isomorphis m i s reduced in 0(n+ra) tim e to rooted tree isomorphism, whic h can be solved in 0(n) time. I t is also possible to find a n implicit representatio n i n linear time. Man y result s on cographs are a con- sequence o f containments i n larger classe s o f graphs, suc h as permutation graphs , distance hereditar y graphs , an d clique-width k graphs, fo r which man y problem s are known to be linear time solvable. Nevertheless , many papers hav e been writte n about complexit y o f problems o n cographs, sinc e so many problem s whic h are dif- ficult o n larger classe s of graphs can be solved efficientl y o n cographs. Example s of tractable problem s o n cographs includ e an 0(n2) max cut algorithm [47] , and lin- ear time algorithms fo r minimum fil l [129] , bandwidth [490] , and pathwidt h (whic h equals treewidt h fo r cographs) [51] . The sandwich proble m fo r cographs i s poly- nomially solvabl e [228] , contrasting wit h the NP-completion o f the completion and deletion problem s fo r cograph s [168] . Othe r NP-complet e problem s o n cograph s include lis t colorin g [279 ] and subgraph isomorphis m [148] .

COMPARABILITY AN D CO-COMPARABILITY GRAPHS : Althoug h a transitiv e orien - tation ca n be found i n linear time , recognizin g comparabilit y graph s i s as hard as recognizing triangle-fre e graphs , currentl y requirin g tim e proportiona l t o matri x multiplication. Th e maximum weighte d cliqu e an d vertex colorin g problem s can be solve d b y a robust algorith m i n linear tim e [367] , and independent se t can be transformed i n linear tim e to bipartite matchin g onc e the transitive orientatio n i s given. Man y problem s o n comparability graph s ar e NP-complete sinc e thes e con - tain the bipartite graphs; for example, Hamilton cycle , dominating set, bandwidth, feedback vertex , treewidt h [24] , minimum fil l [301 ] an d Steiner tre e ar e alread y NP-complete o n bipartite graphs , and bipartite graph s are isomorphism-complete. As far as I can determine, the max cut problem, which is trivial on bipartite graphs, is open fo r comparability graphs . Since [367 ] produces a transitive orientatio n o f the complement i n linear time , maximum independent set and clique cover can be solved robustly on co-comparabil- ity graph s i n linea r time . Hamilto n cycl e ca n be solve d i n 0(n3) time fo r co- comparability graph s [154] . Mor e precisely, the algorithm takes 0(m2) time, wher e i i s the size o f a maximum independen t se t in G; this i s based o n O(z) calls to a Hamilton pat h algorith m fo r co-comparability graph s fro m [149] . Th e authors of [154] als o explai n wh y the algorithm fo r Hamilto n cycl e ca n probably b e imple - mented t o run in 0(n2) time, thoug h th e ful l detail s ar e not worke d ou t in the paper. Minimu m weigh t Steine r tree s i n co-comparabilit y graph s ca n be foun d in 0(nlogn+ra ) tim e [78] . Th e cardinality Steine r tre e ca n be solved i n O(n-f-ra ) time, a s noted independentl y b y Colbourn an d Lubiw; th e algorithm i s described in [333] . Althoug h weighte d dominatin g se t is NP-complete o n co-comparabilit y graphs [93] , the cardinality dominatin g se t problem can be solved i n 0(nm2) tim e [78], an d the problem remain s solvabl e i n polynomial tim e fo r integer weight s up to any fixed k [93] . Note that ther e was a report claimin g a running time fo r domi- nating set proportional to matrix multiplication , but the algorithm wa s apparentl y not correct . Ther e i s an 0(n2ra) algorith m fo r the feedback verte x set problem on co-comparability graph s [347] . Bandwidth , minimu m fill , ma x cut and treewidt h [308, 491 , 47, 24] are NP-complete o n co-bipartite graphs , and co-bipartite graph s are isomorphism-complete . DISK INTERSECTIO N GRAPHS : Recognitio n o f disk intersectio n graph s i s NP-hard [258]. Existenc e an d construction o f an implici t representatio n ar e natural ope n Survey o f Results o n Graph Classe s 309 problems fo r this clas s o f graphs. Anothe r natura l questio n i s whether th e clique problem, whic h i s polynomially solvabl e fo r unit dis k graphs , can be solved fo r the more general class. Dis k graphs contain both unit dis k graphs and planar graphs as subclasses, implyin g NP-completeness o f such problems as 3-coloring, independen t set, clique cover, Steiner tree, feedback vertex, dominating set, Hamilton cycle, an d bandwidth. I f the model is given as input, polynomia l time approximation scheme s for weighted independent set and weighted vertex cover are given in [174]; these also apply to intersection graph s o f regular polygons , in any fixed dimension. I know of no exact algorithmi c result s designe d specificall y fo r disk intersectio n graphs ; thu s problems whic h are solvable on planar graphs , suc h as max cut, minimum fill, and isomorphism, see m t o be open fo r dis k intersectio n graphs . Treewidt h als o doe s not see m to have been studie d fo r disk graphs , and is singled out as an interesting open proble m fo r the subclass o f planar graph s i n [42] .

DISTANCE HEREDITAR Y GRAPHS : Distanc e hereditary graph s ca n be recognized i n linear time , an d an implici t representatio n ca n be constructed i n the same tim e bound. However , a s note d i n [399] , n o natural intersectio n mode l i s know n fo r the class , despite the fact tha t thi s i s an intersection clas s as discussed i n the text. Many problems are solvable efficiently thank s to the pruning sequence, and the fact that thes e hav e clique-widt h a t mos t 3 . Fo r example, maximu m weighte d clique , maximum weight independent set , coloring , and clique cover can be solved in O(n) time i f the pruning sequenc e i s given as input, an d thus i n linear time i f the graph is given in adjacency lis t for m [246] . Linea r time algorithms fo r the dominating set problem ar e given i n [386 , 97]. Weighted Steine r tre e i s solvable i n linear tim e for nonnegative edg e weight s [493] , a s are the treewidth an d minimum fill problem s [82], an d feedback verte x se t [288] . Hamilto n cycl e can be solved i n 0(n2) time on distance hereditary graphs [385] . Isomorphis m ca n be tested in polynomial tim e for distanc e hereditar y graphs , sinc e ever y distanc e hereditar y grap h i s a circl e graph. I n fact, I believe that isomorphis m can be tested in linear time on this class, by reductio n to tree isomorphism , bu t the algorithm onl y appeare d i n a technica l report, and the report doe s not give an explicit time bound [29] , see also [28] . Sinc e every tree is distance hereditary , bandwidt h i s NP-complete fo r the class. Ma x cut is polynomial o n the class sinc e ever y distanc e hereditar y grap h ha s clique-width at mos t 3 , but I know o f no specific algorith m fo r distance hereditar y graphs .

DOMINATION GRAPHS : Th e mos t interestin g proble m fo r dominatio n graph s i s recognition, whic h remains open. Algorithm s on domination graph s generally com e either fro m the definition o f the class, or from the fact that ever y domination grap h is weakly chordal, while NP-completeness results follo w fro m containmen t o f classes such as chordal graphs , trapezoid graph s and tolerance graphs .

DOT PRODUC T k GRAPHS : Th e recognition proble m i s open fo r ever y k > 1 , as is th e proble m o f existenc e an d constructio n o f a n implici t representation . N o algorithmic wor k has been don e fo r k > 1, i.e. threshold graphs .

DOUBLY CHORDA L GRAPHS : Doubl y chorda l graph s ca n be recognize d i n linea r time, sinc e they are exactly equa l to chordal and dually chorda l graphs . Althoug h several algorithm s hav e bee n designe d specificall y fo r doubl y chorda l graphs , the best curren t bound s fo r problems discusse d i n this survey com e fro m result s on the superclasses o f chordal graph s o r dually chorda l graphs . 310 Survey o f Results o n Graph Classe s

DUALLY CHORDA L GRAPHS : Duall y chorda l graph s ca n b e recognize d i n linea r time [67] . Man y problem s ar e NP-complete o n dually chorda l graphs, sinc e addin g a universa l verte x make s an y grap h duall y chordal . Thus , fo r example , clique , independent set , coloring, cliqu e cover , Hamilton cycle , treewidth, an d many othe r problems ca n easily be seen to be NP-complete o n this class . Fo r some problems , adding a universal vertex would make the problem trivial; the paper [67 ] shows that a variet y o f domination an d location problems , includin g unweighte d dominatin g set and Steiner tre e (th e weighted problem s are obviously NP-complete , by adding a universa l verte x wit h appropriat e weight ) ca n be solved i n linear tim e on dually chordal graphs . Th e all pairs shortes t pat h proble m ca n be solved i n 0(n2) tim e on duall y chorda l graph s [68].

EPT GRAPHS : Recognitio n i s NP-complete fo r EPT graphs [227] . Althoug h EPT graphs hav e a n implicit representation , constructin g a n implicit representatio n i n polynomial tim e fro m th e adjacency matri x o f an EPT graph i s an open problem . Every lin e grap h i s an EPT graph, whic h implie s that suc h problem s a s coloring, clique cover , Hamilto n cycle , dominatin g set , an d Steine r tre e ar e NP-complete , and that EPT graph isomorphis m i s isomorphism-complete. Th e coloring proble m on EPT graphs i s equivalent to path colorin g on undirected tre e networks, and has received som e attention; Tarja n [454 ] gave a 3/2 optimal colorin g algorithm, an d a 4/3 optimal algorithm with asymptotic approximatio n rati o 11/1 0 is given in [173]. Weighted cliqu e an d weighted independen t se t are solvable b y robust algorithms , as discusse d i n the text. Othe r problem s no t know n t o be NP-complet e fo r lin e graphs (thes e include max cut, which is polynomially solvabl e fo r line graphs [239] , and bandwidth, whic h is open for line graphs [280]) , seem to be open by way of not having bee n studied . A s noted i n [454] , ever y EPT graph i s decomposable usin g clique separator s int o lin e graphs ; thus , i f a proble m solutio n ca n be constructe d from the solutions on clique-separated components , the complexity on EPT graphs will be the same as the complexity on line graphs. Fo r example, this implies that the complexity o f minimum fill or treewidth o n EPT graphs i s equal to the complexity of these problem s on line graphs.

INTERVAL FILAMEN T GRAPHS : Recognizin g interva l filament graphs , an d creatin g an interva l filament model , ar e open problems . I f the model i s given, cliqu e and independent se t are solvable o n interval filament graph s i n polynomial tim e [211] ; these problem s ar e open fo r robust algorithms .

INTERVAL GRAPHS : Th e first linear tim e algorith m fo r recognizing interva l graph s appeared i n [56] ; many other s hav e appeared , an d on e is describe d i n the text . The origina l recognition algorith m ca n also be used to test isomorphis m o f interval graphs i n linea r tim e [355] . Weighte d independen t set , weighted clique , coloring , and cliqu e cove r ca n be solve d i n O(n ) tim e i f the interval s ar e give n i n sorte d order. Hamilto n cycl e ca n als o b e solve d i n O(n) time give n th e interval model , though thi s i s mor e difficul t [96] . A n O(n ) weighte d dominatin g se t algorith m given i n [92] . Bandwidt h ca n be solve d o n interval graph s i n O(nlogn ) tim e (fo r the decisio n problem ; finding th e optimal bandwidt h take s 0(nlog 2n) tim e usin g this algorithm ) [449] , and weighted feedbac k verte x set in 0(n+ra) tim e [353] . A s far a s I can determine, max cut is open fo r interval graphs, though i t can be solved in linea r tim e fo r unit interva l graph s [50]. Survey o f Result s o n Grap h Classe s 311

INTERVAL NUMBER : Recognizin g graphs with interval number k is NP-complete fo r any fixed k > 1 [481] . N o algorith m i s know n whic h construct s a n implici t repre - sentation o f a graph with interval number 2 in polynomial time. Ever y circular-ar c graph ha s interva l number a t mos t 2 , so bandwidth an d colorin g ar e NP-complet e on th e class . Ever y degre e 3 graph ha s interva l numbe r a t mos t 2 , implyin g NP - completeness fo r suc h problem s a s independen t se t an d Hamilto n cycle , an d con - tainment o f lin e graph s implie s NP-completenes s fo r cliqu e cover , dominatin g set , and Steine r tree , a s wel l a s isomorphism-completeness . Ma x cu t [47 ] is als o NP - complete fo r interval number 2 . Ever y plana r graph has interval number a t mos t 3 [426], implying result s suc h a s NP-completeness o f feedback verte x se t whe n inter - val number i s bounded. Th e real challenge fo r this class is to use the representatio n to solv e problem s effectively ; onl y a fe w papers , suc h a s [494] , whic h give s a 2k times optimal weighte d independen t se t algorith m fo r graph s with interva l numbe r k, attemp t t o do so. Th e cliqu e problem seem s to be open fo r graph s wit h interva l number 2 , and perhap s fo r graph s wit h bounde d interva l numbe r a s well. INVERSION-FREE GRAPHS : Althoug h thes e ar e no t a wel l known clas s o f graphs, I include the m i n this surve y du e t o th e interestin g ope n problems , a s describe d i n the text sectio n o n robust algorithms . Recognitio n o f the clas s i s open. Th e cliqu e problem ca n be solve d i f a model i s given a s input, bu t i s open i f one i s given onl y a promise that th e input i s inversion-free. Solvin g the clique problem with a robust algorithm woul d als o giv e a robus t algorith m fo r findin g a maximu m cliqu e i n a visibility graph . K-POLYGON GRAPHS : I t i s possibl e t o tes t whethe r a grap h i s a /c-polygo n grap h in 0(4 fcn2) time , makin g thi s polynomia l whe n k i s fixed, bu t recognitio n i s NP - complete i f k i s allowed to vary [170] . Sinc e /c-polygo n graphs ar e a subset o f circl e graphs, i t i s possibl e t o construc t a n implici t representatio n i n polynomia l time . In contras t wit h circl e graphs, dominatin g se t (an d a variet y o f other dominatio n problems) ar e polynomial fo r an y fixed k [169] . Natura l problem s t o study includ e those whic h ar e polynomia l fo r permutatio n graphs , bu t ope n o r NP-complet e fo r circle graphs; these include the Hamilton cycle, coloring, and clique cover problems. An approximation algorithm fo r bandwidth with performance ratio 2k2 (th e authors note that a n improvement t o 4/3k2 ha s been communicated t o them) fo r /c-polygo n graphs i s given i n [334] ; I believe i t i s open whether bandwidt h i s NP-complete fo r every fixed k.

PATH GRAPHS : Path graph s ca n be recognized i n 0{n-\-m) tim e [144] ; the natura l intersection mode l i s also produced. Thi s resul t i s not wel l known, sinc e the pape r was published i n a proceedings whic h i s relatively difficul t t o find; the pape r [422 ] gives an O(nra ) tim e algorith m i n a more widel y distributed journal . Ever y pat h graph i s chordal, s o problems suc h a s clique , coloring , independen t set , an d cliqu e cover can be solved in linear time. Dominatin g set [55] , max cut [47] , Hamilton cycle [38], an d bandwidt h ar e NP-complete o n path graphs . Hamilto n cycl e i s also NP- complete on the subclass o f directed path graphs [381 ] and open fo r rooted directe d path graphs ; dominatin g se t i s polynomia l o n roote d directe d pat h graph s [55] . Note that th e math revie w article o f the paper [163 ] is a bit ambiguous , an d migh t lead the reade r (i.e. , I read i t this way ) to believ e dominating se t i s polynomial o n path graphs ; the clas s o f graphs i n the paper i s actually the dually chorda l graphs . Path graph s ar e isomorphism-complet e [54] ; the subclas s o f directe d pat h graph s is als o isomorphism-complete , whil e isomorphis m o f roote d directe d pat h graph s 312 Survey o f Results o n Graph Classe s can b e tested i n polynomial tim e [25] . Path graph s ar e separated fro m spli t and chordal graph s b y the minimum dominatin g cliqu e problem , whic h ca n be solved in polynomial time on path graph s but is NP-complete on split graph s [331] . I was unable to find any work on the Steiner tre e problem restricte d to path graphs , but I a m told tha t thi s shoul d b e NP-complete a s a simpl e xtensio n o f a proo f tha t connected dominatin g se t is NP-complete fo r the class [328] . PERFECT GRAPHS : Recognitio n o f perfect graph s i s a famous ope n problem . Th e difficult, bu t polynomial, algorithm s fo r the clique, coloring, cliqu e cover, and inde- pendent se t [237 ] are robust algorithm s fo r perfect graphs . Perfec t graph s includ e many subclasses , includin g classe s fo r whic h suc h problem s a s Hamilton cycle , dominating set , Steiner tree , feedbac k vertex , bandwidth , treewidth , an d mini- mum fill are NP-complete. Wit h the exception o f the important pape r cite d above, there ar e few results solvin g NP-complet e problem s o n perfect graph s i n general; an 0(n3) combinatorial algorith m fo r 3-coloring on perfect graph s is given in [470]. One area o f algorithmic research i s recognition o f perfect graph s when restricted to various subclasses . Example s includ e recognition o f perfect graph s without certai n induced subgraphs, such as perfect claw-fre e graph s [111] , perfect K± — e-free graph s [197], perfect bull-fre e graph s [405] , paw-free perfec t graph s [389] , and dart-free per - fect graph s [108] . Othe r classe s fo r which perfec t grap h recognitio n i s polynomial include planar graphs [266 ] and 2-split graphs , i.e. graphs which can be partitioned into two split graph s [262] . Ope n problem s i n this are a are collected at www. cs. rutgers. edu/~ch.vatal/perf ect /problems. html.

PERFECTLY ORDERABL E GRAPHS : Recognizin g perfectl y orderabl e graph s i s NP- complete [369] . Ther e are linear time (O(nra) in the weighted case) [259 ] algorithms for solvin g chromati c numbe r an d maximum cliqu e i f a perfec t orderin g i s given as par t o f the input; i f the order i s not given, i t i s open t o solv e thes e problem s without resortin g to more difficult an d general algorithms for perfect graphs . Man y NP-completeness result s follo w fro m th e fact tha t suc h classe s a s chordal graphs , comparability graphs , and other classe s are perfectly orderable . Se e [261] for more information o n perfectly orderabl e graphs . PERMUTATION GRAPHS : Permutatio n graphs can be recognized in linear time [367] , by an algorithm which constructs the natural implicit representation. Independen t set and clique can be solved in O(nloglogn) time if the model is given as input; thes e are equivalent to finding the longest increasin g subsequenc e o f a permutation. Th e well know n algorith m O(nlogn ) fo r finding a longes t increasin g subsequenc e can be modifie d t o run in O(nloglogn ) tim e i f inpu t number s ar e in the range l.. n by usin g vo n Emde Boa s trees , an d ca n als o b e easil y modifie d t o handl e th e weighted case . Sinc e permutatio n graph s ar e perfect, colorin g an d clique cove r can be solved in the same time bound; a maximum cardinalit y matchin g algorith m with th e same runnin g tim e i s given i n [406] . A n 0(n) algorith m fo r finding the minimum cardinalit y dominatin g set problem when given the permutation diagra m is contained in [98]; an 0(n+ra) algorith m for the weighted problem is given in [407]. Isomorphism o f permutation graph s can be tested in 0(n2) time [447] . Steine r tre e can b e solved i n O(n) time, sinc e ever y permutatio n grap h i s a trapezoid graph , and i n O(nlogn ) tim e i n the weighted cas e [275] , i f the permutation diagra m i s given. A linear time algorithm fo r finding a Hamilton cycl e in a permutation grap h is given in [153] . Ther e is an 0(nt) algorith m to determine whethe r a permutation Survey o f Results o n Graph Classe s 313

graph ha s treewidth (whic h equal s pathwidt h fo r permutation graphs ) a t most t [48]. Th e weighted feedbac k verte x se t problem ca n be solved i n O(ran) tim e for permutation graph s [343] . Th e complexity o f bandwidth o n permutation graph s is an ope n problem ; i t is also ope n fo r the subclass o f bipartite permutatio n graphs , but ca n be solved i n polynomial tim e fo r chain graph s [309] . Ma x cut also seem s to be open fo r this clas s of graphs; the the reduction give n in [16] intended to show that the problem i s NP-complete fo r permutation graph s i s not polynomial.

PI GRAPHS : Recognitio n o f PI graphs i n polynomial tim e i s open. Nevertherless , an implici t representatio n ca n be constructed i n polynomial time , sinc e ever y PI graph i s a trapezoid graph . Althoug h ther e hav e bee n paper s writte n solvin g algo - rithmic problem s on PI graphs, I know o f no problem fo r which the best algorith m on PI graphs i s faster tha n the time o f the best curren t know n algorith m fo r trape- zoid graphs . I also kno w o f no results whic h explicitl y separat e the complexity on triangle graph s fro m trapezoi d graphs , and NP-completeness result s com e only as a consequence o f the fact tha t thes e contai n suc h classe s as permutation graph s and interval graphs .

PROBE INTERVA L GRAPHS : Prob e interva l graph s can be recognized i n linear tim e if the divisio n o f the vertices int o probe s and nonprobes i s given [286 , 287], but is open i f the partition i s not given. N o algorithms hav e been specificall y designe d for probe interva l graphs , so any known algorithm s and NP-completeness result s com e only fro m containment s betwee n grap h classes .

SPLIT GRAPHS : Spli t graph s can be recognized i n linear time , and in O(n) time if the degre e sequenc e i s give as input [223] . Spli t graph s hav e the same complexitie s as chorda l graph s o n all problems describe d unde r chorda l graphs , and often see m to b e at the core o f algorithms an d proofs o f difficulty fo r chordal graphs . Th e two classe s are separated i n complexity fo r a number o f problems suc h a s triangle packing, i.e . the problem o f finding the maximum numbe r o f vertex disjoin t trian - gles. Th e triangle packing problem i s polynomial fo r split graphs , but NP-complete for chorda l graph s [239] ; pathwidt h i s also NP-complet e o n chordal graphs , and polynomially solvabl e on split graph s [240] . Th e sandwich proble m (give n a set o f required and optional edges , i s there a set of optional edge s whic h can be included to giv e a graph in the class) is polynomial fo r split graphs , in contrast to most o f the other classe s discusse d here ; specifically , the sandwich proble m i s NP-complete for comparability graphs , permutatio n graphs , co-comparabilit y graphs , circl e graphs , interval graphs , circular-ar c graphs , pat h graphs , chorda l graphs , and co-chordal graphs, an d was left a s an open proble m fo r chordal bipartit e graph s and strongly chordal graph s i n [228] . However , th e split grap h completio n proble m an d split graph deletio n problem s ar e NP-complete [383] , whil e the number o f edges whic h need to be added o r deleted to a given grap h to obtain a split grap h ca n be com- puted i n polynomial tim e [249] . Althoug h bandwidt h i s NP-complete, ther e i s a 2-optimal approximatio n algorith m fo r split graph s [306] .

STRONGLY CHORDA L GRAPHS : Strongl y chorda l graph s ca n b e recognized i n 0(n 2) or O(mlogn ) time . N o implicit representatio n i s known fo r this class . Strongl y chordal graph s ar e both chorda l an d dually chordal , s o such problem s a s inde - pendent set , clique, coloring , dominatin g set , and Steiner tre e ca n be solve d i n linear time . Unlik e bot h o f these classes , weighte d dominatin g se t is polynomia l 314 Survey o f Results o n Grap h Classe s on strongl y chorda l graphs ; th e tim e boun d i s linea r i f the stron g eliminatio n or - dering i s give n a s part o f the inpu t [182] , an d thu s i s 0(n2) o r O(mlogn ) i f give n in adjacenc y lis t form . Th e sam e tim e bound s (wit h eliminatio n orde r give n o r not) appl y t o th e feedbac k verte x se t problem , a s a consequenc e o f [95 ] an d th e correspondence o f feedback verte x se t an d a coverin g problem a s mentioned unde r chordal graphs . Hamilto n cycl e i s NP-complet e o n strongl y chorda l graph s [378] . I believ e that strongl y chorda l graph s ar e isomorphism-complet e b y a simpl e re - duction fro m th e isomorphis m completenes s o f chorda l bipartit e graph s show n i n [380]; note that thi s does not follo w directl y fro m the result o f [25 ] showing that di - rected path grap h isomorphis m i s isomorphism-complete. Thi s paper distinguishe s between roote d directe d pat h graphs , i n whic h al l edg e direction s emanat e fro m the root , an d directe d pat h graphs , i n whic h edge s ca n b e directe d bot h toward s the roo t an d awa y fro m th e root . Th e firs t clas s i s a subclas s o f strongl y chorda l graphs, whil e th e secon d clas s i s no t (i t contain s th e 4-sun) . Unfortunately , bot h classes are sometimes referred t o as directed path graphs, and the paper [282 ] gives directed pat h graph s i n th e firs t sens e a s a subclas s o f strongl y chorda l graphs , leading to potentia l confusion . [25 ] shows that isomorphis m i s polynomial fo r th e rooted case . Bandwidt h i s NP-complete fo r thi s class , sinc e ever y tre e i s strongl y chordal. A s far a s I could determine, max cut i s open fo r strongly chorda l graphs . TOLERANCE GRAPHS : Recognitio n o f toleranc e graph s i s a n ope n problem , a s i s existence o f a n implici t representation . Althoug h i t i s not obvious , recognitio n o f tolerance graph s i s in NP [255] . Mos t algorithm s o n tolerance graph s ar e a conse - quence o f containmen t i n weakl y chorda l graphs , bu t a fe w algorithm s hav e bee n designed t o wor k o n tolerance graph s whe n a representation i s given a s part o f the input. Fo r example , the cliqu e an d colorin g problem s ca n b e solve d i n 0(n2) tim e on tolerance graphs, i f the toleranc e representatio n i s given [232] . Independen t se t and cliqu e cover can be solved in linear time given a tolerance representation, usin g the fac t tha t a co-perfect orderin g comes directly fro m the tolerance representatio n [229], and that independen t se t an d cliqu e cove r ca n be solve d i n linear tim e give n a co-perfec t orderin g [109] . A n 0(n 5) algorith m fo r minimu m fill o n th e clas s o f multitolerance graphs, which includes both tolerance and trapezoid graphs, i s given in [393] . Bandwidt h i s NP-complet e fo r toleranc e graph s [272] ; thi s i s the onl y example I know o f a problem prove d to be NP-complete o n tolerance graphs whic h is not know n t o b e NP-complet e o n a subclass . Man y algorithmi c problem s see m to be open fo r toleranc e graphs, whether th e mode l i s given o r not. I might choose dominating se t an d Hamilto n cycl e a s interestin g examples , bot h wit h th e mode l given and fo r robus t algorithms ; these problems are solvable i n polynomial time o n bounded toleranc e graph s an d NP-complet e o n weakly chorda l graphs . TRAPEZOID GRAPHS : Trapezoi d graph s ca n b e recognize d i n 0(n2) tim e [357] , b y an algorith m whic h construct s th e natura l implici t intersectio n model . Trapezoi d graphs are co-comparability graphs, but a number o f algorithms have been designed to solv e problem s mor e efficientl y tha n o n th e large r class . Give n th e model , th e chromatic number , cliqu e cover , weighte d independen t set , an d weighte d cliqu e problems ca n b e solve d i n O(nloglogn ) tim e [188] ; /c-colorin g ca n als o b e solve d in 0(kn) tim e [140] . I t i s als o possibl e t o solv e the Steine r tre e proble m i n O(n ) time [344 ] give n the trapezoida l representation . Minimu m weigh t dominatin g set s can b e foun d i n O(nra ) tim e [345] , an d th e treewidt h an d minimu m fil l problem s can b e solve d i n 0(n 2) tim e [49] . I n general , algorithm s fo r trapezoi d graph s Survey o f Result s o n Grap h Classe s 315 received les s attentio n tha n algorithm s fo r man y o f th e othe r grap h classes . I found n o specifi c paper s dealin g wit h Hamilto n cycle , unweighte d domination , o r isomorphism whe n restricte d t o trapezoi d graphs ; i n al l thes e cases , I fee l tha t behavior o f trapezoid graphs should be closer to the subclass o f permutation graph s than the superclass o f co-comparability graphs . I n particular, havin g worked on the related recognitio n proble m fo r trapezoi d graphs , I fee l that ther e i s a polynomia l time isomorphis m algorithm , thoug h non e ha s ye t bee n developed . Alon g thes e lines, [190 ] remark s tha t th e O(nra ) algorith m fo r solvin g th e weighte d feedbac k vertex set problem on permutation graph s [343 ] can easily be extended to trapezoi d graphs. Althoug h [308 ] give s a 2-approximatio n algorith m fo r trapezoi d graph s which i s simpler an d faste r tha n th e algorith m fo r AT-fre e graphs , the complexit y of bandwidt h o n trapezoi d graph s seem s t o b e a n ope n problem ; ma x cu t i s als o apparently open . TREEWIDTH k: Althoug h computing the treewidth i s NP-complete, i t i s possible to determine whethe r a graph ha s treewidth a t mos t k fo r fixed k i n linear tim e [43] , and t o find th e correspondin g tre e decomposition . Ever y proble m whic h ca n b e posed i n second order monadic logi c can be solved fo r graphs o f bounded treewidt h [132]; the clas s o f problem s solvabl e i s extended i n [59] , and algorithm s hav e als o been designed fo r many problems which do not fit in a general framework o f solvable problems. Ther e are too many problem s which hav e been solve d o n graphs o f fixed treewidth t o dea l wit h here ; [44 ] is a surve y devote d t o thi s issue . Problem s suc h as clique, independent set , traveling salesman, an d dominating se t ca n be solved i n O(n) tim e on graphs o f bounded treewidth. Ma x cut i s O(n) o n graphs o f bounded treewidth [47] ; note tha t eve n th e weighte d cas e i s solvabl e i n linea r time , whil e weighted ma x cu t i s NP-complet e o n an y clas s whic h contain s arbitraril y larg e cliques. Isomorpis m ca n be tested i n polynomial time fo r graph s o f fixed treewidth /c, thoug h k appear s i n th e exponen t o f th e runnin g tim e [45] . Bandwidt h i s on e of the bette r know n problem s whic h i s NP-complete o n the class , sinc e i t remain s NP-complete fo r trees . UNIT DIS K GRAPHS : Uni t dis k grap h recognitio n i s NP-hard [77] ; membership i n NP i s open . Existenc e an d constructio n o f a n implici t representatio n i s a n ope n problem, /c-colorin g fo r ever y k > 2 i s NP-complet e fo r uni t dis k graph s [234] , as i s independen t se t [112] . Dominatin g set , Hamilto n cycle , bandwidth , an d Steiner tre e ar e NP-complet e fo r th e subclas s o f gri d graph s [112 , 278 , 157 , 207]. Clique i s solvabl e b y a robus t algorith m i n polynomia l time , a s discusse d i n th e text. Polynomia l time approximatio n scheme s (thes e achiev e approximation ratio s of 1-f- e fo r an y e , with running time polynomial i n the siz e o f the input an d 1/e ) fo r independent set , vertex cover and dominating se t o n unit dis k graphs are presente d in [273] ; extension s fo r independen t se t an d verte x cove r ar e discusse d unde r dis k graphs. Goo d approximatio n algorithm s fo r bandwidt h an d variou s othe r verte x ordering problem s o n intersectio n graph s o f randoml y generate d uni t disk s ar e presented i n [156] . [362 ] give s a colorin g algorith m wit h approximatio n rati o 3 for uni t dis k graphs ; thi s ca n b e improve d t o a ratio o f 2 i f the cliqu e siz e i s fixed and a mode l i s give n a s par t o f the inpu t [233] . Ther e i s a considerabl e bod y o f work dealin g wit h unit-dis k graph s a s a n idealize d mode l fo r radi o transmission , solving broadcasting problem s o f various types ; there i s als o a body o f work usin g unit dis k graph s fo r ma p labeling . Sinc e I a m no t exper t i n these areas , I will no t try t o pic k ou t th e importan t result s o n these subjects . Althoug h ther e i s a grea t 316 Survey o f Results o n Graph Classe s deal o f work dealin g with algorithm s fo r specialized problem s o n unit dis k graphs , the complexit y o f many o f the classical graph problem s o n unit dis k graph s seem s to be open. Thus , I was not able to determine the complexity o f such problem s as clique cover, isomorphism, treewidth, minimum fill, feedback verte x set, or max cut when restricte d t o unit dis k graphs , whethe r th e model wa s given a s part o f the input o r not. Interested readers should note that uni t dis k graphs are studied unde r many differen t name s (on e common nam e i s geometric graphs ) i n the literature. VISIBILITY GRAPHS : Visibilit y grap h recognitio n i s a wel l know n ope n problem , both i n th e cas e whe n a n orderin g o f vertice s o n the outsid e o f the polygo n i s specified, an d when any ordering is permissible. N o implicit representation i s known for thi s clas s o f graphs. I f the model i s given, the clique problem ca n be solved in polynomial tim e fo r visibilit y graphs ; thi s i s ope n fo r robus t algorithms , an d is discussed i n the text. Independen t se t an d dominatin g se t ar e NP-complet e fo r visibility graphs , an d isomorphis m i s as hard a s fo r genera l graph s [349] . Cliqu e cover i s also NP-complet e fo r visibility graph s [137] ; a logarithmic approximatio n algorithm i s given i n [167] . Th e complexity o f coloring visibilit y graph s i s an open problem. Althoug h ever y visibility grap h ha s a Hamilton cycle , it i s open whethe r you ca n find a Hamilto n cycl e i n a visibilit y grap h whe n th e inpu t i s give n i n adjacency matri x form . WEAKLY CHORDA L GRAPHS : Weakl y chorda l graph s ca n b e recognize d i n 0(ra 2) time usin g tw o fundamentall y differen t algorithm s [36 , 256], both o f whic h ar e discussed i n the text. Th e best know n algorithms fo r independent set , clique cover , coloring, an d cliqu e ru n in O(nra ) tim e fo r the unweighted cas e [256 ] and 0(n4) time fo r the weighted cas e [446] . NP-complet e problem s includ e dominatin g set , max cut, Steiner tree , bandwidth, an d Hamilton cycle , whic h are NP-complete on chordal graphs . Treewidt h an d minimu m fill-in ca n be solve d i n 0(n6) tim e o n weakly chorda l graph s [64] . Feedbac k verte x doe s no t see m to hav e bee n studie d for this class. Weakl y chordal graphs seem much more general than chordal graphs, but ther e are relatively fe w problems know n to be tractable o n chordal graph s but NP-hard o n weakly chorda l graphs ; th e most natura l proble m I could find o f this form i s testing whether a n input grap h i s perfectly orderabl e [260] , which i s trivial on chordal graphs . WELL COVERE D GRAPHS : Well-covere d graph s ar e co-NP-complete t o recognize . As i s show n i n the text , th e independen t se t problem , whic h i s trivia l o n well - covered graph s give n a promis e tha t th e inpu t i s i n the class , i s intractabl e fo r robust algorithms . Man y problems, such as clique, coloring, Steine r tree, minimu m fill, ma x cut , Hamilto n cycle , cliqu e cover , an d dominating se t ar e NP-complet e on thi s class , an d well-covered graph s ar e isomorphism-complete, al l of which are shown in [419]. I t is easy to show that other problems are NP-complete, since adding pendant vertice s adjacen t t o eac h verte x o f G make s an y graph well-covered ; fo r example, it is easy to see that treewidt h and minimum fill are NP-complete i n this way. Althoug h I would not classify i t as a significant ope n problem, it is interesting to not e tha t thi s transformatio n canno t b e used t o sho w tha t bandwidt h i s NP- complete. Therefore , despit e the fac t tha t bandwidt h i s NP-complete o n most o f the classes above and most optimizatio n problem s are hard fo r well-covered graphs , bandwidth ha s not been studie d fo r this clas s and cannot b e trivially show n to be NP-complete. Distinction s betwee n complexit y o f problems on well-covered graph s and problem s on subclasses suc h as very well-covered graph s (i n which all maximal Survey o f Result s o n Grap h Classe s 317 independent set s have size n/2, and fo r whic h recognition, cliqu e cover, dominatin g set, an d Hamilto n cycl e are polynomial) ar e studied i n [418] . This page intentionally left blank Bibliography

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M-join, 174 , 17 5 bipartite graph , 88 , 106 , 126 , 170 , 30 8 S-ordering, 12 9 bipartite permutatio n graph , 88 , 96 , 126 , T-free, 96 , 18 3 133 r-free matrix , 111 , 112, 120 , 124 , 132 , 14 5 bisimplicial edge , 130 , 13 3 Q, 6 bitonic path , 11 5 6, 6 blocking vertex , 25 5 /c-clique extendible ordering , 24 3 Boland, 8 6 /c-closure, 66 , 7 0 Bondy, 6 6 fc-module, 16 6 Booth, 145 , 19 3 /c-polygon graph , 130 , 223, 31 1 Borie, 51 , 163, 164 , 25 3 /c-simplicial vertex , 18 5 bottleneck Mong e matrix , 12 6 /c-tree, 5 0 Bouchet, 212 , 21 5 2-join, 171 , 174, 17 5 Bouchitte, 3 4 bounded substitutio n decompositio n diame - Abbas, 12 9 ter, 160 , 17 8 adjacency list , 9 bounded toleranc e graph , 58 , 248, 254, 30 5 adjacency matrix , 8 , 9 boxicity, 36 , 47, 53, 222, 223, 233, 262, 30 5 adjacent, 6 Brandstadt, 20 , 18 5 Agarwal, 7 4 brittle graph , 189 , 25 3 Alekseev, 10 6 Buckingham, 30 6 Alon, 54 , 55 , 7 4 Buneman, 26 7 alternance graph , 21 2 angle order , 58 , 22 4 Cai, 24 8 Anstee, 14 3 Cameron, 25 1 arboricity, 14 6 Cenek, 4 2 Aronov, 7 4 certificate, 8 7 art galler y problem , 8 3 chain cover , 21 0 asteroidal triple , 85 , 129 , 13 4 chain cove r problem , 20 8 asteroidal triple-fre e graph , 85 , 86 , 88 , 90 , 93-95 chain dimension , 20 8 AT-free comparabilit y graph , 9 5 chain graph , 125 , 134 , 20 8 AT-free graph , 85 , 88 , 91 , 93 , 94 , 96 , 102 , characterization problem , 1 4 249, 30 4 Chazelle, 75 , 7 9 autograph, 2 8 Chen, 12 9 Chepoi, 18 5 Bl-orientable, 4 7 chord, 3 1 balanced /c-module , 166 , 17 9 chordal bipartit e graph , 88-90 , 94 , 95 , 111, balanced ske w partition , 17 4 112, 114 , 118-126 , 130 , 132 , 135 , 136 , Bandelt, 20 , 18 7 159, 183 , 203, 211, 255, 306, 31 3 bandwidth, 305-31 6 chordal comparabilit y graph , 33 , 60, 62 , 64, bar-representable graph , 8 0 69, 70 , 9 0 Benzer, 11 , 12, 26 5 chordal graph , 29 , 31, 33, 38, 47, 49, 50, 60, Berge, 13 6 61, 63-65 , 70 , 71 , 85-88 , 90 , 95 , 111 , Berge graph, 50 , 25 1 120-122, 124 , 133 , 187 , 189 , 191 , 196 , biconvex graph , 96 , 128 , 13 4 224, 225 , 246 , 257 , 262 , 267 , 272 , 305 , bipartite complement , 131 , 208 312, 31 3

337 338 Index chromatic number , 8 , 36 , 40, 49 , 8 0 convex graph , 96 , 126 , 128 , 133 , 145 , 254 , Chvatal, 46 , 66 , 83, 24 5 306 circle graph, 20 , 39 , 42, 47, 49-51, 130 , 134 , convex ordering , 12 8 159, 187 , 212 , 228 , 306, 311, 313 Corneil, 39 , 51, 82, 98, 19 3 circle intersectio n graph , 3 6 cotree, 98 , 10 9 circle order , 42 , 54 , 58 , 224, 26 2 Coullard, 8 2 circle trapezoid graph , 30 6 counting problem , 1 3 circular permutation graph , 39 , 42, 217, 218, covering graph , 22 3 228 Cunningham, 213 , 21 4 circular-arc filamen t graph , 4 4 curve contac t graph , 25 4 circular-arc graph , 35 , 38 , 47 , 48 , 123 , 124 , cycle-free partia l order , 6 0 149, 159 , 194 , 197 , 202 , 211 , 225 , 237 , 241, 252 , 260 , 269 , 272 , 273 , 306 , 307 , Dahlhaus, 10 , 11 7 311, 31 3 decision problem , 7 circular-arc overla p graph , 24 8 deterministically decomposable , 16 1 Clark, 239 , 24 0 diameter, 189 , 22 5 claw-free graph , 103 , 104, 109, 110, 191, 237, diametral pair , 22 5 312 Dilworth, 10 7 clique, 8 , 32-34, 36-38, 40-42, 44, 46-49, 80, Dilworth number , 10 7 169, 233 , 235 , 239 , 241 , 242 , 247 , 248 , dimension, 41 , 49, 50 , 61, 94, 107 , 118 , 133, 253, 304-31 7 176, 177 , 222 , 229 , 249, 26 2 clique cover , 8 , 46 , 305-307 , 309-312 , 314 , Dirac, 19 1 316, 31 7 directed pat h graph , 311 , 314 clique cutset , 167 , 23 4 disk graph , 24 9 clique separator, 167 , 169 , 170 , 179 , 23 3 disk intersectio n graph , 36 , 48 , 53 , 55 , 57 , clique separato r decomposition , 22 1 223, 262 , 30 8 clique tree , 31, 64, 25 8 distance hereditary graph , 20 , 161, 165, 181, clique-width, 164-166 , 178 , 247, 307, 30 9 182, 187 , 191 , 308, 30 9 clone, 92 , 10 9 distance labeling , 25 , 2 9 co-AT-free graph , 91 , 93 dominating pair , 249 , 30 4 co-bipartite edg e eliminatio n order , 24 0 dominating set , 36 , 48, 70, 80, 185 , 248, 249, co-bipartite graph , 106 , 17 0 304-317 co-chordal graph , 87 , 90 , 94 , 95 , 196 , 224 , domination graph , 50 , 58 , 133 , 309 313 dot produc t graph , 56 , 58, 30 9 co-comparability graph , 33 , 38 , 39 , 47 , 49 , doubly chorda l graph , 185 , 187 , 188 , 191 , 51, 60 , 86, 88, 90, 95, 305, 308, 313, 314 234, 30 9 co-interval graph , 19 6 doubly conve x graph , 128 , 14 5 co-planar graph , 4 1 doubly lexica l ordering , 112 , 120 , 121 , 132 , Cogis, 124 , 20 9 133, 14 6 cograph, 28 , 97, 109, 160, 165, 178, 188, 191, Dragan, 18 5 307 dually chorda l graph , 124 , 185 , 186 , 188 , Colbourn, 239 , 24 0 191, 234 , 310, 311, 313 coloring, 8 , 32 , 35 , 36 , 38 , 46 , 87 , 169 , 245 , Dushnick, 41 , 87 248, 304-31 7 dynamic programming , 3 4 comparability graph , 28 , 29 , 31 , 33, 38 , 50 , 60, 64 , 70 , 88 , 90 , 91 , 94, 96 , 102 , 107 , edge, 6 110, 123 , 124 , 157 , 159 , 176 , 177 , 179 , Edmonds, 25 1 191, 196 , 209 , 212 , 217 , 235 , 243 , 246 , Edwards, 26 9 251, 253 , 308, 31 3 elimination, 18 1 complement, 6 enclosure property , 13 3 completely decomposable , 16 1 envelope, 7 7 composition sequence , 44 , 5 0 EPT graph , 38 , 59 , 69 , 102 , 167 , 220 , 222 , connected matrix , 138 , 24 9 233, 234 , 252 , 262 , 269 , 31 0 consecutive I s property , 14 5 Erdos, 19 4 construction problem , 1 3 Eschen, 19 7 containment model , 5 1 Euler, 8 5 convex, 13 3 Everett, 8 2 convex fan , 76 , 24 4 external visibilit y graph , 74 , 8 3 Index 339 f-diagram, 5 1 induced subgraph , 6 Foldes, 8 7 induced visibilit y graph , 76 , 79 , 83, 25 3 Fiiredi, 14 3 intersection characterization , 5 0 feedback vertex , 305-31 5 intersection class , 9 6 Fellows, 4 0 intersection graph , 5 Felsner, 5 8 intersection model , 4 4 Ferrers digraph , 20 9 interval dimension , 94 , 124 , 209 , 212 , 222 , Ferrers dimension , 124 , 20 9 227 fill, 167 , 17 9 interval filament graph , 40 , 44, 49, 224 , 24 8 Ford, 8 9 interval graph, 11 , 16, 17 , 27, 33-35, 42, 47- fraternally orientable , 4 7 50, 56 , 60, 85, 86, 88, 90, 129 , 145 , 193, Fulkerson, 89 , 20 5 196, 197 , 204 , 259 , 265 , 269 , 273 , 275 , 307, 310 , 31 3 Gabor, 21 2 interval number , 37 , 48, 223, 249, 262 , 31 1 Gallai, 8 8 interval order , 196 , 209, 22 7 Gamarnik, 26 9 inversion-free, 24 4 Garey, 65 , 167 , 22 2 inversion-free graph , 255 , 31 1 Gavril, 40 , 42, 108 , 22 0 inversion-free ordering , 25 4 generalized implici t representation , 22 , 23 , isomorphism, 80 , 305-312, 314-31 7 79, 117 , 13 2 isomorphism complete , 5 9 generalized join , 17 8 Ghosh, 8 1 Jamison, 23 3 Gilmore, 8 6 Jegou, 2 8 Golumbic, 5 , 20 , 51 , 58, 60 , 65 , 85 , 87 , 90 , Johnson, 65 , 167 , 222, 239 , 24 0 130, 191 , 212, 233, 26 5 join decomposition , 160 , 182 , 188 , 212, 213, Grotschel, 14 , 4 6 228 graph, 6 graph drawing , 5 k-join, 22 8 greedy coloring , 47 , 51, 246, 25 2 k-polygon graph , 4 0 grid, 38 , 4 8 k-sparse graph , 1 9 grid graph , 38 , 305, 31 5 k-tree, 27 , 27 2 gridicity, 12 9 Kamula, 3 9 Gross, 20 5 Kannan, 17 , 19 , 21, 22 Gustedt, 10 , 15 9 Karapetjan, 26 9 Kierstead, 6 3 Habib, 2 8 Klee, 8 3 Halin graph , 163 , 16 4 Kloks, 10 3 Hamilton cycle , 8 , 40 , 87 , 241, 304-317 Kolen, 12 1 Hammer, 8 7 Kratochvil, 40 , 41, 222, 24 7 Hay ward, 5 8 Kratsch, 10 3 Helly circular-ar c graph , 20 7 Kubena, 4 1 Helly property , 33 , 197 , 23 3 hereditary, 19 , 48, 76 , 10 6 labeled graph , 6 Hiraguchi poset , 4 9 LB-simplicial, 21 8 Hoang, 18 0 LBFS, 70 , 168 , 188 , 192 , 225, 26 3 Hoahg, 17 9 Le, 2 0 Hoffman, 86 , 12 1 Lehel, 11 9 hole, 46 , 5 0 Lekkerkerker, 8 6 homogeneous set , 15 0 lexicographic breadt h first search , 94 , 188 , Hsu, 212 , 269 , 27 5 189, 192 , 194 , 24 9 Lin, 74 , 8 0 implicit grap h conjecture , 1 9 line graph, 27 , 170 , 221, 234, 305 , 310, 31 1 implicit grap h question , 8 5 line o f sigh t graph , 8 0 implicit representation , 17 , 19-21 , 25 , 28 , linear extension , 32 , 41, 15 7 33, 37 , 48 , 53 , 73 , 98 , 110 , 134 , 160 , linear matrix , 136 , 14 6 161, 166 , 25 3 local complement , 215 , 22 8 independent set , 8 , 33, 35-37, 40, 43, 44, 46, local structure , 1 8 49, 50 , 80, 110 , 163 , 169 , 233, 237, 238, Lovasz, 14 , 4 6 249, 250 , 252 , 304-31 7 Lubiw, 82 , 120 , 12 1 340 Index

Lueker, 145 , 168 , 19 3 partial order , 9 6 partially complemented , 1 0 Miiller, 10 3 partitionable graph , 4 9 Ma, 169 , 21 2 partitioning, 149-151 , 157 , 159 , 17 7 Maffray, 17 9 path graph , 21 , 28, 38, 59, 68, 220, 269, 311, matching, 70 , 189 , 237 , 253, 306, 307 , 31 2 313 Matousek, 4 0 path ordering , 9 5 matrix multiplication , 86 , 101 , 103, 10 9 pathwidth, 308 , 31 3 max cut , 305 , 307-311, 313-31 7 pendant vertex , 20 , 18 1 maximal, 8 perfect elimination , 6 4 maximal cardinalit y search , 18 8 perfect eliminatio n bipartit e graph , 65 , 130 , McConnell, 10 , 115 , 149 , 155 , 158 , 18 0 133 MCS, 18 8 perfect eliminatio n order , 6 4 Miller, 41 , 87 perfect eliminatio n scheme, 31, 111, 121, 124, minimal fill, 167 , 16 9 191 minimal separator , 34 , 48, 49, 111 , 189, 21 8 perfect graph , 32 , 46 , 49 , 50 , 69 , 87 , 107 , minimal separators , 3 6 170, 173 , 228, 246, 31 2 minimum fill, 91 , 167 , 306 , 308-310 , 312 , perfectly orderable graph, 120 , 245, 253, 255, 314, 316 , 31 7 312, 31 6 Minty, 10 3 Perl, 9 8 modular decomposition , 92 , 160 , 19 4 permutation graph , 38 , 47-49 , 51 , 58 , 60 , modulator, 24 8 87, 88 , 90 , 95 , 96 , 98 , 109 , 126 , 149 , module, 92 , 109 , 150 , 152 , 175 , 17 6 158, 160 , 178 , 191 , 198 , 217 , 306 , 308 , Monge matrix , 142 , 14 6 311-313 Monma, 22 0 PI graph , 39 , 224 , 31 3 monotone property , 9 PI* graph , 39 , 22 4 monotonic path , 11 5 planar graph , 20 , 38, 69, 177 , 309, 311, 312 Moore, 21 1 planar poset , 22 3 Mulder, 20 , 18 7 polygon-circle graph , 41 , 51 Muller, 17 , 18 , 21, 22 PQ-tree, 129 , 145 , 193 , 197 , 199 , 20 6 multitolerance graph , 31 4 PQ-trees, 14 5 prime module , 15 3 N-free poset , 2 7 probe interva l graph , 58 , 266, 31 3 Naji, 21 2 projective plane , 13 6 Nakajima, 4 2 proper circular-ar c graph , 205 , 22 7 Naor, 17 , 19 , 21, 22 proper interva l graph , 20 4 Nastos, 9 5 proximity graph , 3 6 neighbor, 6 pruning sequence , 18 1 neighborhood containment , 123 , 13 3 pseudoline, 8 1 Nesetril, 24 7 NLC-width, 164 , 17 8 Qin, 6 3 nondeterministically decomposable , 16 1 quotient graph , 92 , 15 3 nonredundant edge , 25 4 NP-complete, 7 Raghavan, 66 , 27 4 Ramsey theory , 2 4 O, 6 realizer, 15 8 Olariu, 19 3 recognition problem , 1 3 optimization problem , 1 4 reconstructible vertex , 9 2 order notation , 6 rectangle number , 118 , 13 3 ordered chorda l graph , 82 , 8 3 rectangle overla p graph , 24 8 overlap, 42 , 47, 49, 5 1 recursively decomposabl e graph , 163 , 17 8 overlap graph , 21 2 reduced matrix , 11 3 overlap model , 5 1 redundant 1 , 11 3 representation problem , 1 3 Paige, 12 1 Rim, 4 2 Papoutsakis, 26 3 Roberts, 26 5 parallel module , 15 3 robust algorithm , 231 , 250, 316 , 31 7 Parker, 163 , 16 4 robust NP-complete , 25 4 partial fc-tree, 27 , 16 3 robust-NP-complete, 25 0 Index 341 rooted directe d pat h graph , 59 , 312, 31 4 Todinca, 3 4 Rose, 16 8 tolerance graph , 57 , 58, 134 , 247, 31 4 Rudich, 17 , 19 , 21, 22 topological sort , 3 2 total graph , 2 8 Sakarovich, 12 1 total interva l number , 37 , 48, 22 3 sandwich problem , 308 , 31 3 totally balance d hypergraph , 11 9 Sbihi, 10 3 Tovey, 163 , 16 4 Schaffer, 30 7 trampoline, 18 2 Scheinerman, 23 , 44, 51, 55, 10 5 transitive closure , 6 0 Schnyder, 22 2 transitive graph , 15 8 Schrijver, 14 , 4 6 transitive orientation , 31 , 51, 150 , 156-158 , semiorder, 17 8 176, 19 7 series module , 15 3 transitive reduction , 60 , 17 6 series-parallel graph , 15 , 38, 164 , 27 2 trapezoid graph , 38 , 50 , 58 , 94 , 124 , 130 , Shamir, 5 8 134, 209 , 211 , 227 , 248 , 254 , 305 , 309 , Shepherd, 4 9 312-314 Shih, 269 , 27 5 traveling salesman , 31 5 shortest path , 5 1 tree, 51 , 17 8 sign pattern, 5 4 treewidth, 91 , 272, 305 , 306 , 308-310 , 312 - simple eliminatio n order , 70 , 182 , 185 , 18 9 317 simple matrix , 14 3 Trenk, 5 8 simple vertex , 169 , 18 2 triangle, 15 8 simplicial vertex , 31 , 102, 107 , 109 , 179 , 19 1 triangle-extendible graph , 244 , 25 3 skew partition, 170 , 171 , 173, 17 9 triangle-extendible ordering , 24 2 Skiena, 74 , 8 0 triangle-free graph , 49 , 100 , 101 , 103 , 109 , source, 156 , 17 6 110, 19 1 sphericity, 3 7 triangulation, 27 2 spider graph , 41 , 49 Trotter, 41 , 63, 21 1 Spinrad, 20 , 197 , 21 2 Tucker, 197-199 , 203 , 205, 22 5 split, 21 2 Turan, 10 9 split graph , 21 , 29, 60, 87, 90, 104 , 106 , 189 , twin, 20 , 24 , 18 1 262, 305 , 312, 31 3 two dimensional partial order, 41, 48, 60, 87, Sritharan, 21 8 158, 176 , 191 , 202, 210 , 21 8 stable set , 8 two-pair, 67 , 69, 70 , 17 9 star cutset , 171 , 17 4 Stathopolous, 1 6 uniformly sparse , 1 9 Steiner tree , 305-31 7 unique neighbo r set , 9 2 Steinhaus, 6 6 unit circular-ar c graph , 206 , 226 , 26 0 Steinhaus graph , 65 , 6 9 unit dis k graph , 36 , 239, 309 , 31 5 Sterten, 17 9 unit interva l graph , 178 , 204, 259 , 26 3 Stewart, 42 , 98, 129 , 193 , 306 unit-cost assumption , 8 Stobert, 15 , 6 9 universal graph , 21 , 22 string graph , 40 , 22 3 strong ordering , 12 7 vertex, 6 strong perfec t grap h conjecture , 17 0 vertex cover , 253 , 31 5 strongly chorda l graph , 112 , 118 , 119 , 124 , vertex expansion , 44 , 5 0 169, 182 , 185 , 189 , 191 , 234, 31 3 vertex multiplication , 4 5 subgraph isomorphism , 30 8 visibility graph , 55 , 73 , 74 , 76 , 80 , 81 , 83, substitution decomposition, 98 , 149, 152, 156, 126, 241 , 243-245 , 253 , 255 , 262 , 311 , 176-178, 21 2 316 sun, 18 2 Supowit, 21 2 Warren, 5 4 Suri, 7 4 Warren represen t able, 5 5 Warren's theorem , 54-56 , 73 , 74, 7 7 Tarjan, 121 , 149, 167-169 , 220 , 23 3 weak order , 17 7 Tarski, 5 7 weakly chordal graph, 32 , 67, 69, 71, 83, 87- Thorup, 27 2 91, 93-96 , 110 , 123 , 124 , 179 , 218 , 219 , threshold graph , 56 , 58 , 107 , 30 9 306, 309 , 314, 31 6 threshold signe d graph , 10 7 weakly triangulate d graph , 32 , 19 1 342

Wei, 22 0 well-covered graph , 232 , 250 , 252 , 31 7 Whitesides, 167 , 169 , 17 9 width,107

Yannakakis, 65 , 167 , 22 2 Yu, 12 9

Zito, 23 , 10 5 Titles i n Thi s Serie s

19 Jerem y P . Spinrad , Efficien t grap h representations , 200 3 18 Olav i Nevanlinna , Meromorphi c function s an d linea r algebra , 200 3 17 Vital y I . Voloshin , Colorin g mixed hypergraphs : theory, algorithm s an d applications , 200 2 16 Nea l Madras , Lecture s o n Mont e Carl o Methods , 200 2 15 Brad d Har t an d Matthe w Valeriote , Editors , Lecture s o n algebrai c mode l theory , 2002 14 Fran k de n Hollander , Larg e deviations , 200 0 13 B . V . Rajaram a Bhat , Georg e A . Elliott , an d Pete r A . Fillmore , Editors , Lectures i n operator theory , 200 0 12 Salm a Kuhlmann , Ordere d exponentia l fields, 200 0 11 Tibo r Krisztin , Hans-Ott o Walther , an d Jianhon g Wu , Shape , smoothnes s an d invariant stratificatio n o f an attractin g se t fo r delaye d monoton e positiv e feedback , 199 9 10 Jif l Patera , Editor , Quasicrystal s an d discret e geometry , 199 8 9 Pau l Selick , Introductio n t o homotop y theory , 199 7 8 Terr y A . Loring , Liftin g solution s to perturbin g problem s i n C*-algebras , 199 7 7 S . O . Kochman , Bordism , stabl e homotop y an d Adam s spectra l sequences , 199 6 6 Kennet h R . Davidson , C*-Algebra s b y example, 199 6 5 A . Weiss , Multiplicativ e Galoi s modul e structure , 199 6 4 Gerar d Besson , Joachi m Lohkamp , Pierr e Pansu , an d Pete r Peterse n Miroslav Lovric , Maun g Min-Oo , an d McKenzi e Y.-K . Wang , Editors , Riemannian geometry , 199 6 3 Albrech t Bottcher , Aa d Dijksm a an d Hein z Langer , Michae l A . Dritsche l an d James Rovnyak , an d M . A . Kaashoe k Peter Lancaster , Editor , Lecture s o n operato r theor y an d it s applications , 199 6 2 Victo r P . Snaith , Galoi s modul e structure , 199 4 1 Stephe n Wiggins , Globa l dynamics , phas e spac e transport, orbit s homoclini c t o resonances, an d applications , 199 3