Table Of ContentDevelopments in Mathematics
Teresa W. Haynes
Stephen T. Hedetniemi
Michael A. Henning Editors
Topics in
Domination
in Graphs
Developments in Mathematics
Volume 64
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
Topics in Domination
in Graphs
Editors
TeresaW.Haynes StephenT.Hedetniemi
DepartmentofMathematicsandStatistics SchoolofComputing
EastTennesseeStateUniversity ClemsonUniversity
JohnsonCity,TN,USA Clemson,SC,USA
DepartmentofMathematicsandApplied MichaelA.Henning
Mathematics DepartmentofMathematicsandApplied
UniversityofJohannesburg Mathematics
Johannesburg,SouthAfrica UniversityofJohannesburg
Johannesburg,SouthAfrica
ISSN1389-2177 ISSN2197-795X (electronic)
DevelopmentsinMathematics
ISBN978-3-030-51116-6 ISBN978-3-030-51117-3 (eBook)
https://doi.org/10.1007/978-3-030-51117-3
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Whileconceptsrelatedtodominationingraphscanbetracedbacktothemid-1800s
in connection with various chessboard problems, domination was first defined
as a graph-theoretical concept in 1958. Domination in graphs experienced rapid
growthsinceitsintroductionresultinginover1200paperspublishedondomination
in graphs by the late 1990s. 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
GraphsandDominationinGraphs:AdvancedTopics.WerefertotheseasBooksI
andII.
Theexplosivegrowthhascontinuedandtodaymorethan4000papershavebeen
publishedondominationingraphs,andthematerialinBooksIandIIismorethan
20 years old. Thus, the authors think it is time for an update on the developments
indominationtheorysince1998.Wealsowanttogiveacomprehensivetreatment
ofthemajortopicsindomination.Thiscoverageofdominationincludingboththe
fundamentalmajorresultsandupdateswillbeintheformofthreebooks,whichwe
shallcallBooksIII,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
results that illustrate many of the proof techniques that are used in domination
theory. For the companion books, Books IV and V, we invited leading researchers
indominationtocontributechapters.
Book V has three parts, the first of which focuses on several domination-
related concepts. The second part focuses on domination in (i) hypergraphs, (ii)
chessboards, and (iii) digraphs and tournaments. The third part focuses on the
developmentofalgorithmsandcomplexityofdominationparameters.
Thepresentvolume,BookIV,concentratesonmajordominationparametersthat
were not covered in Book III. Although well over 70 types of dominating sets
have been defined, Book IV focuses on the primary ones that have received the
most attention in the literature. In particular, the chapters include such parameters
v
vi Preface
as paired domination, connected domination, restrained domination, domination
functions,Romandomination,andpowerdomination.
The authors of Book IV provide a survey of known results with a sampling of
prooftechniquesforeachparameter.Toavoidexcessiverepetitionofdefinitionsand
notations,Chapter1providesaglossaryofcommonlyusedtermsandChapter gives
anoverviewofmodelsofdominationfromwhichtheparametersaredefined.
This book is intended as a reference resource for researchers and is written
to reach the following audience: First, the audience includes the established
researchersinthefieldofdominationwhowantanupdatedcomprehensivecoverage
on domination. Second are the researchers in graph theory and graduate students
whowishtobecomeacquaintedwithtopicsindominationincludingmajoraccom-
plishmentsinthefieldandprooftechniquesused.Weanticipatethatitcouldalsobe
usedinaseminarcourseondominationingraphs.
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
ModelsofDominationinGraphs............................................... 13
TeresaW.Haynes,StephenT.Hedetniemi,andMichaelA.Henning
PairedDominationinGraphs .................................................. 31
WyattJ.Desormeaux,TeresaW.Haynes,andMichaelA.Henning
ConnectedDomination .......................................................... 79
MustaphaChellaliandOdileFavaron
RestrainedandTotalRestrainedDominationinGraphs.................... 129
JohannesH.HattinghandErnstJ.Joubert
MultipleDomination............................................................. 151
AdrianaHansbergandLutzVolkmann
DistanceDominationinGraphs................................................ 205
MichaelA.Henning
Locating-DominationandIdentification ...................................... 251
AntoineLobstein,OlivierHudry,andIrèneCharon
SignedandMinusDominatingFunctionsinGraphs ........................ 301
LiyingKangandErfangShan
FractionalDominatingParameters............................................ 349
WayneGoddardandMichaelA.Henning
RomanDominationinGraphs.................................................. 365
MustaphaChellali,NaderJafariRad,SeyedMahmoudSheikholeslami,
andLutzVolkmann
RainbowDominationinGraphs................................................ 411
BoštjanBrešar
vii
viii Contents
EternalandSecureDominationinGraphs.................................... 445
WilliamF.KlostermeyerandC.M.Mynhardt
StratifiedDomination............................................................ 479
GaryChartrandandPingZhang
GlobalDomination............................................................... 497
RobertC.Brigham,JulieR.Carrington,andRonaldD.Dutton
PowerDominationinGraphs................................................... 521
PaulDorbec
Glossary of Common Terms
TeresaW.Haynes,StephenT.Hedetniemi,andMichaelA.Henning
1 Introduction
Itisdifficulttosaywhenthestudyofdominationingraphsbegan,butforthesakeof
thisglossaryletussaythatitbeganin1962withthepublicationbyOysteinOre’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,
had previously defined the same concept, but had, in Chapter 4 The Fundamental
T.W.Haynes((cid:2))
DepartmentofMathematicsandStatistics,EastTennesseeStateUniversity,JohnsonCity,TN
37614-0002,USA
DepartmentofMathematicsandAppliedMathematics,UniversityofJohannesburg,Auckland
Park2006,SouthAfrica
e-mail:[email protected]
S.T.Hedetniemi
ProfessorEmeritus:SchoolofComputing,ClemsonUniversity,Clemson,SC29634,USA
e-mail:[email protected]
M.A.Henning
DepartmentofMathematicsandAppliedMathematics,UniversityofJohannesburg,Auckland
Park2006,SouthAfrica
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2020 1
T.W.Haynesetal.(eds.),TopicsinDominationinGraphs,Developmentsin
Mathematics64,https://doi.org/10.1007/978-3-030-51117-3_1
2 T.W.Haynesetal.
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.
de Jaenisch in 1862 [7], Franz Nauck in 1850 [14], and Max Bezzel in 1848 [4]
fortheircontributionstothesetypesofchessboardproblemsinvolvingdominating
setsofchesspieces.
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 defi-
nitions. 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,andmatchingnumbersaredefinedinSection3.3.
Finally in Section 3.4, we define selected domination-type parameters that will
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.
