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Developments in Mathematics Teresa W. Haynes Stephen T. Hedetniemi Michael A. Henning Editors Structures of Domination in Graphs Developments in Mathematics Volume 66 SeriesEditors KrishnaswamiAlladi,DepartmentofMathematics,UniversityofFlorida, Gainesville,FL,USA PhamHuuTiep,DepartmentofMathematics,RutgersUniversity,Piscataway,NJ, USA LoringW.Tu,DepartmentofMathematics,TuftsUniversity,Medford,MA,USA AimsandScope The Developments in Mathematics (DEVM) book series is devoted to publishing well-writtenmonographswithinthebroadspectrumofpureandappliedmathemat- ics.Ideally,eachbookshouldbeself-containedandfairlycomprehensiveintreating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High-quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVMseriesappealstoavarietyofaudiencesincludingresearchers,postdocs,and advancedgraduatestudents. Moreinformationaboutthisseriesathttp://www.springer.com/series/5834 Teresa W. Haynes • Stephen T. Hedetniemi Michael A. Henning Editors Structures of Domination in Graphs 123 Editors TeresaW.Haynes StephenT.Hedetniemi DepartmentofMathematicsandStatistics SchoolofComputing EastTennesseeStateUniversity ClemsonUniversity JohnsonCity,TN,USA Clemson,SC,USA DepartmentofMathematics andAppliedMathematics UniversityofJohannesburg Johannesburg,SouthAfrica MichaelA.Henning DepartmentofMathematics andAppliedMathematics UniversityofJohannesburg Johannesburg,SouthAfrica ISSN1389-2177 ISSN2197-795X (electronic) DevelopmentsinMathematics ISBN978-3-030-58891-5 ISBN978-3-030-58892-2 (eBook) https://doi.org/10.1007/978-3-030-58892-2 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicenseto SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface While concepts related to domination in graphs can be traced back to the mid- 1800s in connection to various chessboard problems, domination was first defined asagraphtheoreticalconceptin1958.Dominationingraphshasexperiencedrapid growthfromitsintroduction,resultinginover1200paperspublishedondomination ingraphsbythelate1990s. Noting the need for a comprehensive survey of the literature on domination in graphs, in 1998 Haynes, Hedetniemi, and Slater published the first two books on domination, Fundamentals of Domination in Graphs and Domination in Graphs: AdvancedTopics.WerefertothesebooksasBooksIandII. The explosive growth of this field since 1998 has continued, and today more than4,000papershavebeenpublishedondominationingraphs,andthematerialin BooksIandIIisnowmorethan20yearsold.Thus,theauthorsfeelitistimeforan updateonthedevelopmentsindominationtheorysince1998.Wealsowanttogive a comprehensive treatment of only the major topics in domination. This coverage ofdomination,includingthemajorresultsandupdates,willbeintheformofthree books,whichwecallBooksIII,IV,andV. Book III, Domination in Graphs: Core Concepts, is written by the authors and concentrates, as the title suggests, on the three main types of domination in graphs: domination, independent domination, and total domination. It contains major results on these basic domination numbers, including proofs of selected resultsthatillustratemanyoftheprooftechniquesusedindominationtheory. For the companion books, Books IV and V, we invited leading researchers in dominationtocontributechapters. Book IV concentrates on the most-studied types of domination that are not covered in Book III. Although well over 70 types of domination have been defined, Book IV focuses on those that have received the most attention in the literature, and contains chapters on paired domination, connected domination, restraineddomination,multipledomination,distancedomination,dominatingfunc- tions, fractional dominating parameters, Roman domination, rainbow domination, locating-domination, eternal and secure domination, global domination, stratified domination,andpowerdomination. v vi Preface The present volume, Book V, is divided into three parts. The first part focuses on several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings, irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination, and spectral graph theory. The secondpartcoversdominationin(i)hypergraphs,(ii)chessboards,and(iii)digraphs and tournaments. The third part focuses on the development of algorithms and complexity of (i) signed, minus, and majority domination, (ii) power domination, and(iii)alliancesingraphs.Thethirdpartalsoincludesachapteronself-stabilizing dominationalgorithms. The authors of the chapters in Book V provide a survey of known results with a sampling of proof techniques in their areas of expertise. To avoid excessive repetition of definitions and notation, Chapter 1 provides a glossary of commonly usedterms. This book is intended as a reference resource for researchers and is written to reach the following audiences: first, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; second, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and third, graduate students with interests in graph theory,whomightfindthetheoryandmanyreal-worldapplicationsofdomination of interest for master’s and doctoral theses topics. We also believe that Book V provides a good focus for use in a seminar on either domination theory or domination algorithms and complexity, including the new algorithm paradigm of self-stabilizingdominationalgorithms. Wewishtothanktheauthorswhocontributedchapterstothisbookaswellasthe reviewersofthechapters. JohnsonCity,TN,USA TeresaW.Haynes Clemson,SC,USA StephenT.Hedetniemi Johannesburg,SouthAfrica MichaelA.Henning Contents GlossaryofCommonTerms .................................................... 1 TeresaW.Haynes,StephenT.Hedetniemi,andMichaelA.Henning PartI RelatedParameters BroadcastDominationinGraphs .............................................. 15 MichaelA.Henning,GaryMacGillivray,andFeiranYang AlliancesandRelatedDominationParameters............................... 47 TeresaW.HaynesandStephenT.Hedetniemi FractionalDomatic,Idomatic,andTotalDomaticNumbers ofaGraph......................................................................... 79 WayneGoddardandMichaelA.Henning DominatorandTotalDominatorColoringsinGraphs ...................... 101 MichaelA.Henning Irredundance ..................................................................... 135 C.M.MynhardtandA.Roux ThePrivateNeighborConcept ................................................. 183 StephenT.Hedetniemi,AliceA.McRae,andRaghuveerMohan AnIntroductiontoGameDominationinGraphs............................ 219 MichaelA.Henning DominationandSpectralGraphTheory...................................... 245 CarlosHoppen,DavidP.Jacobs,andVilmarTrevisan VarietiesofRomanDomination................................................ 273 M.Chellali,N.JafariRad,S.M.Sheikholeslami,andL.Volkmann vii viii Contents PartII DominationinSelectedGraphFamilies DominationandTotalDominationinHypergraphs.......................... 311 MichaelA.HenningandAndersYeo DominationinChessboards..................................................... 341 JasonT.HedetniemiandStephenT.Hedetniemi DominationinDigraphs......................................................... 387 TeresaW.Haynes,StephenT.Hedetniemi,andMichaelA.Henning PartIII AlgorithmsandComplexity AlgorithmsandComplexityofSigned,Minus,andMajority Domination ....................................................................... 431 StephenT.Hedetniemi,AliceA.McRae,andRaghuveerMohan AlgorithmsandComplexityofPowerDominationinGraphs.............. 461 StephenT.Hedetniemi,AliceA.McRae,andRaghuveerMohan Self-StabilizingDominationAlgorithms....................................... 485 StephenT.Hedetniemi AlgorithmsandComplexityofAlliancesinGraphs ......................... 521 StephenT.Hedetniemi Glossary of Common Terms TeresaW.Haynes,StephenT.Hedetniemi,andMichaelA.Henning 1 Introduction Itisdifficulttosaywhenthestudyofdominationingraphsbegan,butforthesake ofthisglossaryletussaythatitbeganin1962withthepublicationofOysteinOre’s book Theory of Graphs [15]. In Chapter 13 Dominating Sets, Covering Sets and IndependentSetsof[15],weseeforthefirsttimethenamedominatingset,defined asfollows:“AsubsetDofV isadominatingsetforGwheneveryvertexnotinD istheendpointofsomeedgefromavertexinD.”Orethendefinesthedomination number, denoted δ(G), of a graph G, as “the smallest number of vertices in any minimaldominatingset.”So,atthispoint,andforthefirsttime,dominationhasa “name”anda“number.” Of course, prior to this Claude Berge [3], in his book Theory of Graphs and its Applications, which was first published in France in 1958 by Dunod, Paris, T.W.Haynes((cid:2)) DepartmentofMathematicsandStatistics,EastTennesseeStateUniversity,JohnsonCity, TN,USA DepartmentofMathematicsandAppliedMathematics,UniversityofJohannesburg, Johannesburg,SouthAfrica e-mail:[email protected] S.T.Hedetniemi SchoolofComputing,ClemsonUniversity,Clemson,SC,USA e-mail:[email protected] M.A.Henning DepartmentofMathematicsandAppliedMathematics,UniversityofJohannesburg, Johannesburg,SouthAfrica e-mail:[email protected] ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1 T.W.Haynesetal.(eds.),StructuresofDominationinGraphs,Developments inMathematics66,https://doi.org/10.1007/978-3-030-58892-2_1 2 T.W.Haynesetal. had previously defined the same concept, but had, in Chapter 4 The Fundamental Numbers of the Theory of Graphs of [3], given it the name “the coefficient of externalstability.” Before Berge, Dénes König, in his 1936 book Theorie der Endlichen und Unendlichen Graphen [13], had defined essentially the same concept, but in VII Kapitel,BasisproblemfürgerichteteGraphen,Königgaveitthename“punktbasis,” whichwewouldtodaysayisanindependentdominatingset. AndevenbeforeKönig,inthebooksbyDudeneyin1908[8]andW.W.Rouse Ballin1905[2],onecanfindtheconceptsofdomination,independentdomination, and total domination discussed in connection with various chessboard problems. AnditwasBallwho,inturn,creditedsuchpeopleasW.Ahrensin1910[1],C.F. deJaenischin1862[7],FranzNauckin1850[14],andMaxBezzelin1848[4]for theircontributionstothesetypesofchessboardproblemsinvolvingdominatingsets ofchesspieces. ButitwasOrewhogavethenamedominationandthisnametookroot.Notlong thereafter,CockayneandHedetniemi[6]gavethenotationγ(G)forthedomination numberofagraph,andthisalsotookrootandisthenotationadoptedhere. Sincethesubsequentchaptersinthisbookwilldealwithdominationparameters, there will be much overlap in the terminology and notation used. One purpose of this chapter is to present definitions common to many of the chapters in order to prevent termsbeingdefined repeatedly andtoavoidotherredundancy. Also,since graph theoryterminology and notationsometimes vary,inthisglossaryweclarify theterminologythatwillbeadoptedinsubsequentchapters. We proceed as follows. In Section 2.1, we present basic graph theory definitions. We discuss common types of graphs in Section 2.2. Some fundamental graph constructions are given in Section 2.3. In Section 3.1 and Section 3.2, we presentparametersrelatedtoconnectivityanddistanceingraphs,respectively.The covering, packing, independence, and matching numbers are defined in Section 3.3.FinallyinSection3.4,wedefineselecteddomination-typeparametersthatwill occurfrequentlythroughoutthebook. For more details and terminology, the reader is referred to the two books FundamentalsofDominationinGraphs[10]andDominationinGraphs,Advanced Topics [11], written and edited by Haynes, Hedetniemi, and Slater and the book TotalDominationinGraphsbyHenningandYeo[12].Anannotatedglossary,from which many of the definitions in this chapter are taken, was produced by Gera, Haynes,Hedetniemi,andHenningin2018[9]. 2 BasicTerminology Inthissection,wegivebasicdefinitions,commontypesofgraphs,andfundamental graphconstructions.