2-domination and annihilation numbers

44 Pages·2015·0.5 MB·English

TThhee UUnniivveerrssiittyy ooff SSoouutthheerrnn MMiissssiissssiippppii
TThhee AAqquuiillaa DDiiggiittaall CCoommmmuunniittyy
Honors Theses Honors College
Spring 5-2015
22--DDoommiinnaattiioonn aanndd AAnnnniihhiillaattiioonn NNuummbbeerrss
Sean C. Patterson
University of Southern Mississippi
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TheUniversityofSouthernMississippi
2-DOMINATIONANDANNIHILATIONNUMBERS
by
SeanCristPatterson
AThesis
SubmittedtotheHonorsCollegeof
TheUniversityofSouthernMississippi
inPartialFulfillment
oftheRequirementsfortheDegreeof
BachelorofScience
intheDepartmentofMathematics
May2015
ii
Approvedby
SamuelLyle,Ph.D.,ThesisAdviser
AssociateProfessor
BerndSchroeder,Ph.D.,Chair
DepartmentofMathematics
EllenWeinauer,Ph.D.,Dean,
HonorsCollege
iii
ABSTRACT
Using information provided by Ryan Pepper and Ermelinda DeLaVina in their paper On
the 2-Domination number and Annihilation Number, I developed a new bound on the 2-
domination number of trees. An original bound, γ (G)≤ n+n1, had been shown by many
2 2
other authors. Our goal was to generate a tighter bound in some cases and work towards
generatingamoregeneralboundonthe2-dominationnumberforallgraphs. Throughoutthe
spanofthisprojectIgeneratedandprovedtheboundγ (T)≤ 1(n+2n +n ). Toprovethis
2 3 1 2
boundIfirstprovedthatthe2-dominationnumberofatreewaslessthanorequaltothesum
oftwosub-treesformedbythedeletionofanedge: γ (T)≤γ (T )+γ (T ). Fromthere,I
2 2 1 2 2
provedourboundbyshowingthataminimumcounter-exampledidnotexist. Alargeportion
oftheresultsinvolvescaseswhereagraphT isconsideredtheminimumcounter-example
forthesakeofcontradiction. Fromthere,IshowedthatifT wasacounter-example,thena
sub-treeT was alsoacounter-example,meaning thatT would nolongerbe theminimum
1
counter-example. The last portion of the results is a section comparing the two bounds
onthe2-dominationnumber withrespecttothenumberof edgesandthedegreesof those
edges.
KeyWords: Annihilationnumberofagraph,2-Dominationnumberofagraph,trees
iv
ACKNOWLEDGMENTS
IextendallofthethanksintheworldtomyadviserDr. Lyle. Withouthispatienceand
incrediblemathematicalskillsguidingthisproject,Ifearnothingwouldhavebeendone. I
alsowanttothanktheUniversityofSouthernMississippiMathDepartmentforpreparing
metotakeonsuchanendeavorandtheHonorsCollegeforencouragingmetodoso.
v
TABLE OF CONTENTS
LISTOFILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 AlternateProofofγ (T)≤ n+n1 11
2 2
4.2 GeneratingaNewFormula: 1(n+2n +n ) 13
3 1 2
4.3 Provingγ (T)≤γ (T )+γ (T ) 14
2 2 1 2 2
4.4 Minimizingtheboundsonthecounter-exampleforγ (T)≤ 1(n+2n +n ) 14
2 3 1 2
4.5 ComparingFormulas: γ (T)≤ n+n1 andγ (T)≤ n+2n1+n2 31
2 2 2 3
5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vi
LIST OF ILLUSTRATIONS
Figure
1.1 EvenCycle,C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
8
1.2 A2-DominatingSetofMinimumOrderfortheGraphinFigure1.1 . . . . . . 3
1.3 ReferenceforAnnihilationNumberDiscussionwithDegreesMarked . . . . . 4
2.1 ExamplesoftheChromaticNumberofaGraph . . . . . . . . . . . . . . . . . 6
2.2 Exampleofa2-DominatingSet . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1 AlternatingsetS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 AnnihilationNumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 SplittingaTreeExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 GraphT: MaximumPathwith4Leaves . . . . . . . . . . . . . . . . . . . . . 15
4.5 Graph: T : MaximumPathwith4LeavesMinus2Leaves . . . . . . . . . . . . 15
1
4.6 GraphT: MaximumPathP,Vertexx withdegree3 . . . . . . . . . . . . . . . 17
4.7 GraphT : MaximumPathwith3LeavesMinusOneLeaf . . . . . . . . . . . . 17
1
4.8 MaximumPathwith2Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.9 MaximumPathwith2Leavesin2-DominationSet . . . . . . . . . . . . . . . 19
4.10 GraphT: MaximumPathwith2Leaveswherethedegreeofuis2 . . . . . . . 19
4.11 GraphT : MaximumPathwith2Leaveswherethedegreeofuis2 . . . . . . . 20
1
4.12 GraphT: MaximumPathwith2Leaveswhered(u)=3 . . . . . . . . . . . . 20
4.13 GraphT : MaximumPathwith2Leaveswhered(u)=3withSegmentRemoved 20
1
4.14 GraphT: MaximumPathwith2Leaveswhered(u)>3 . . . . . . . . . . . . 21
4.15 Graph T : Maximum Path with 2 Leaves where the degree of d(u) > 3 and
1
segmentremoved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.16 MaximumPathwith1Leafwherethedegreeofuis1 . . . . . . . . . . . . . . 23
4.17 MaximumPathwith1Leafwherethedegreeofuis2 . . . . . . . . . . . . . . 23
4.18 Maximum Path with 1 Leaf where the degree of u is 2 and x and y have been
removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.19 Maximum Path with 1 Leaf where the degree of u is 3, x and y are removed,
anduisinsetS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.20 MaximumPathwith1Leafwherethedegreeofuis3anduisinsetS forT . . 25
1
4.21 MaximumPathwith1Leafwherethedegreeofuis3anduisnotinsetS for
T . T andT arebothdisplayed. Oneofthedependentpathsisoflength2. . . . 26
1 1
4.22 MaximumPathwith1Leafwherethedegreeofuis3anduisnotinsetS for
T . T andT arebothdisplayed. Thedependentpathisaleaf. . . . . . . . . . . 27
1 1
4.23 MaximumPathwith1Leafwherethedegreeofuisgreaterthan3anduisin
setS forT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1
vii
4.24 MaximumPathwith1Leafwherethedegreeofuisgreaterthan3anduisin
setS forT . x andyareremoved. . . . . . . . . . . . . . . . . . . . . . . . . . 28
1
4.25 Maximumpathwith1Leafwherethedegreeofuisgreaterthan3anduisnot
insetS forT . T andT arebothdisplayed. Thedependentpathsareallleaves. 29
1 1
4.26 MaximumPathwith1 Leafwherethedegreeofuisgreaterthan 3anduisnot
in set S for T . T and T are both displayed. One of the dependent paths is of
1 1
length2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.27 SampleTreeT withn=11,n =6,n =4,andγ (T)=7 . . . . . . . . . . . 31
1 1 2 2
4.28 SampleTreeT withn=17,n =9,n =1,andγ (T)=12 . . . . . . . . . . 32
2 1 2 2
viii
1
Chapter 1
Introduction
Ermelinda DeLaVina and Ryan Pepper wrote an article [DelavinaandPepper,2008] in
2008 entitled "2-Domination and Annihilation Number" which describes results from a
computer program named Graffiti.pc. The paper presents a conjecture which suggests
thatthe2-dominationnumberofagraphG,denotedbyγ (G),islessthanorequaltothe
2
annihilation number of a graph, denoted by a(G), plus one [DelavinaandPepper,2008].
Precisely,
γ (G)≤a(G)+1.
2
A significant amount of progress has been made by such authors as Wyatt Desormeaux,
MichaelHenning,DouglasRall,andAndersYeowhenGisatree[Desormeauxetal.,2014].
Thisresearchprojectattemptstoexpandupontheirworkandtheworkofmanyothersby
analyzinga boundonthe 2-dominationnumbersuggested bythis conjecture andprovinga
specialcase. Indoingso,thisprojecthelpedtoexpandthefieldofgraphtheorywhichhas
variedpracticalapplicationsintheareasofbiosystematics, socialnetworkinganalysis, and
topology.
According to DeLaVina, Graffiti.pc is a "conjecture-making computer program" that
hasbeenusedtohelpsupport,form,anddiscreditpredictionsmadeprimarilyinthefields
ofgraph theoryandchemistry [DeLaVina,2002]. Itis usedtosearch throughmanygraphs
andmakeconjecturesabouttherelationshipsbetweengraphparameters. Graffiti.pcisthe
secondgenerationofthiscomputerprogram. Theoriginalversion,Graffiti,wascreatedby
SiemionFajtlowiczandlaterenhancedbyDeLaVinato makeamoreuserfriendlyversion
[DeLaVina,2005].
Thisprojectisinthefieldofmathematicscalledgraphtheory. Agraph,inthecontextof
thisproject,isasetofverticesandasetofedges. AgraphisoftendenotedbyG=(V,E),
where E is the set of edges, and V is the set of vertices. The edge set is a subset of the
orderedpairsofV×V. Inthisthesis,onlysimple,looplessgraphsareconsidered;thatis,no
vertexisinanedgewithitselfandtherearenorepeatededges. Twoverticesareconsidered
adjacent if there is an edge between them. For a subset S of vertices in the graph to be