Table Of ContentDevelopments in Mathematics
Teresa W. Haynes
Stephen T. Hedetniemi
Michael A. Henning Editors
Structures
of Domination
in Graphs
Developments in Mathematics
Volume 66
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KrishnaswamiAlladi,DepartmentofMathematics,UniversityofFlorida,
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PhamHuuTiep,DepartmentofMathematics,RutgersUniversity,Piscataway,NJ,
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LoringW.Tu,DepartmentofMathematics,TuftsUniversity,Medford,MA,USA
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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)
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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 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, 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.
