Words and Graphs PDF
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Monographs in Theoretical Computer Science An EATCS Series Sergey Kitaev Vadim Lozin Words and Graphs Monographs in Theoretical Computer Science An EATCS Series Editors: M. Henzinger J. Hromkovi(cid:254) M. Nielsen G. Rozenberg A. Salomaa Founding Editors:W.Brauer G.Rozenberg A.Salomaa OnbehalfoftheEuropeanAssociation forTheoreticalComputerScience(EATCS) AdvisoryBoard: S. Albers H. Attiya G. Ausiello M. Broy C. Calude A. Condon A. Czumaj P. Degano J. Diaz P. Gastin G. Gottlob D. Harel J. Hartmanis R. Heckel L.A. Hemaspaandra T. Henzinger M. Hermenegildo B. Jonsson J. Karhumäki L. Kari M. Koutny D. Kozen T. Leighton H. Lin G. Mauri M. Nivat D. Niwi(cid:276)ski C. Papadimitriou D. Peleg D. Sannella U. Schöning D. Scott P.G. Spirakis D. Wagner E. Welzl M. Wirsing More information about this series at http://www.springer.com/series/776 Sergey Kitaev • Vadim Lozin Words and Graphs Sergey Kitaev Vadim Lozin Department of Computer Mathematics Institute and Information Sciences University of Warwick University of Strathclyde Coventry, UK Glasgow, UK SeriesEditors Monika Henzinger Juraj Hromkovi(cid:254) Faculty of Science ETHZentrum Universität Wien DepartmentofComputerScience Wien, Austria SwissFederalInstituteofTechnology Zürich,Switzerland Mogens Nielsen Department of Computer Science Grzegorz Rozenberg Aarhus Universitet Leiden Centre of Advanced Aarhus, Denmark Computer Science Leiden University Arto Salomaa Leiden, The Netherlands Turku Centre of Computer Science Turku, Finland ISSN 1431-2654 ISSN 2193-2069 (electronic) Monographs in Theoretical Computer Science. An EATCS Series ISBN 978-3-319-25857-7 ISBN 978-3-319-25859-1 (eBook) DOI 10.1007/978-3-319-25859-1 Library of Congress Control Number: 2015955140 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Foreword This excellent book by Sergey Kitaev and Vadim Lozin is the first book devoted to theimportanttopicofword-representablegraphs. Itisacomprehensivepresentation ofthestateoftheart,gathering,unifyingandsummarizingthelargebodyofresults thathavebeendevelopedoverrecentyears. Itraisesnewopenproblemstobefurther explored by researchers, and then introduces additional connections between words and graphs, which are worth studying further. Letw beawordoveranalphabetV andsupposethatxandy aretwodistinct letters in w. We say that x and y alternate in w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy... (of even or odd length) or a wordyxyx...(ofevenoroddlength). Ifxandy donotalternateinw, wesaythat these letters are non-alternating in w. A graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x,y)∈E for each x(cid:3)=y. Thefamilyofword-representablegraphsgeneralizesseveralwellknownclasses including comparability graphs, 3-colorable graphs, circle graphs and graphs of ver- tex degree at most 3. But not all graphs are word-representable, making them a family of interest. Indeed, the authors suggest a variety of problems and directions for further research, some open for several years and others published in this book for the first time. The book offers much more than an introduction to the theory of word- representablegraphs. Itpresentsacollectionofvariousinterrelationsbetweenwords andgraphsintheliterature. Theseincludediscussionofwellknownnotionssuchas Pru¨fer sequences, Gray codes or de Bruijn graphs, and some less well known rela- tionsbetweengraphsandwords,suchaspermutationgraphs,polygon-circlegraphs, word-digraphs and path-schemes. In conclusion, “Words and Graphs” is the most valuable and definitive source volumeforenteringthisinterestingareaofresearch. Itisenjoyabletoread,andwill become very useful for both learning and reference. Its challenging problems give insight into the progression of research over the last decade, and will keep a new v vi Foreword generation well occupied for the coming decade. Haifa, June 2015 Martin Charles Golumbic Preface In 1918, Heinz Pru¨fer [120] discovered a fascinating relationship between labelled trees with n vertices and sequences of length n−2 made of the elements of the set {1,2,...,n}. This relationship is, in fact, a bijection, i.e. a one-to-one correspon- dencebetweentreesand sequences, and it allowedPru¨ferto proveCayley’sformula about the number of n-vertex labelled trees. The Pru¨fer sequence is a classical ex- ample showing the importance of words for graph enumeration. More importantly, with the advent of the computer era representing graphs by words became crucial for storing graphs in computer memory. Words have also been used to reveal and describe various useful properties of graphs, such as classes where many difficult algorithmic problems become easy, or classes that are well-quasi-ordered by the in- ducedsubgraphrelation. Ontheotherhand,graphshavefrequentlybeenexploited to study various properties of words and related combinatorial structures, such as permutations. Since the discovery of the Pru¨fer sequence, the interplay between words and graphs has repeatedly been investigated in both directions. One of the most recent findings in this area is the notion of word-representable graphs. This is a common generalization of several well-studied classes of graphs, such as circle graphs, com- parability graphs, 3-colourable graphs and graphs of degree at most 3 (also known as subcubic graphs). The invention of word-representable graphs became the inspi- ration for writing this book. On the other hand, it motivated us to look at various other important relationships between words and graphs. In the first part of the book, wereport, inacomprehensiveway, thestateoftheartonword-representable graphsandgiveabrieftouroverrelatedgraphclasses. Inthesecondpart,weexplore many other connections between words and graphs. In no way is our description of these connections comprehensive or complete. Rather, it is an invitation to an area that faces many great challenges and offers the prospect of many great discoveries. The book is organized as follows. • InChapter1weprovideaquickintroductiontotheworldofword-representable graphs, which is the main object of our interest in this book, and give a few vii viii Preface motivating points for our study of these graphs. • InChapter2wediscusshereditaryclassesofgraphs. Thereasonforourrather thoroughdiscussionisthattheclassofword-representablegraphsishereditary anditgeneralizesseveralimportantrepresentativesofthisfamily. Thus,know- ing general resultsand approachesto tackleproblemsfor hereditary classesof graphs can be useful in dealing with word-representable graphs. In particu- lar, the only enumerative result for word-representable graphs known to date was obtained by means of asymptotic enumeration developed for hereditary classes. • In Chapters 3–5 we provide a comprehensive introduction to, and overview of known results in, the theory of word-representable graphs. A proof of each result is presented along with a reference to the original source. • In Chapter 6 we discuss a generalization of the notion of word-representable graphs, namely that of u-representable graphs, where u is a binary word over {1,2}. Our word-representable graphs are 11-representable in the new termi- nology. Thefocusofthechapteristhestudyof12-representablegraphs,where many interesting properties of these graphs are established; all the proofs are provided in a self-contained manner. We also show that any graph is u-representable assuming that u is of length at least 3. • In Chapter 7 we suggest a variety of problems and directions for further re- searchinthetheoryofword-representablegraphs. Theserangefromproblems open for several years to those published in this book for the first time. The chapteralsocontainsasectiondedicatedtopossibleapproachestotackleprob- lems on word-representable graphs. • In Chapters 8 and 9 we discuss interrelations of words and graphs in the literature by means other than word-representability. This includes a variety of topics without a common thread. We discuss not only well known notions, such as Pru¨fer sequences, Gray codes or de Bruijn graphs, but also less well known objects, such as permutation graphs and polygon-circle graphs, and structuresessentiallyunknowntothegeneralaudience,suchasword-digraphs and path-schemes. • Appendix A contains, normally standard, definitions in graph theory that are used in the book, while in Appendix B we provide a few basic definitions in algebra,analysisandcombinatoricsthatareusedinthebook. Notethatmany other (graph-theoretic) definitions are incorporated in other chapters, usually at the places where we need them. Preface ix Chapters 2–5 contain exercises and solutions to selected problems, making thisbooksuitableforteachingpurposes. Thebookwillbeofinteresttoresearchers, graduate students and advanced undergraduate students with interests in graph theory, combinatorics (on words) and discrete mathematics. Acknowledgments The main focus of this book is the theory of word-representable graphs, and we wouldliketothankalltheresearcherscontributingtothedevelopmentofthetheory. Special thanks go to Magnu´s M. Halld´orsson and Artem Pyatkin for coming up with key results in the area, which influenced directions of further research in the field. It should also be noted that the theory would not be possible had Steven Seif not approached the first author of the book with a combinatorial problem in algebra, which resulted in the appearance of the prototype of the notion of a word- representable graph. Also, we are thankful to the people who worked on creating software to work with word-representable graphs. These are O¨zgu¨r Akgu¨n, Herman Z.Q. Chen, Ian Gent, Marc Glen, Christopher Jefferson, Alexander Konovalov and Steve Linton. Further, we would like to thank Manda Riehl for proofreading the first two chapters in the initial draft of the book, and providing many useful comments. ThefirstauthorwouldliketoexpresshisgratitudetoIanRuthvenforallocat- ing extra time to work on this book. He is also grateful to his family for constant support, especially to his sons, Daniel and Nicholas Kitaev, for inspiration. The second author is grateful to his wife, Irina Lozina, for encouragement, both by words and graphs, as graphs are nothing but relations. Glasgow, September 2015 Sergey Kitaev Coventry, September 2015 Vadim Lozin
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