Algorithmic Aspects of Graph Connectivity (Encyclopedia of Mathematics and its Applications)

392 Pages·1970·4.2 MB·English

by  Hiroshi Nagamochi

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AlgorithmicAspectsofGraphConnectivity
AlgorithmicAspectsofGraphConnectivityisthefirstbookthatthoroughlydiscusses
graphconnectivity,acentralnotioningraphandnetworktheory,emphasizingitsal-
gorithmic aspects. This book contains various definitions of connectivity, including
edge-connectivity,vertex-connectivity,andtheirramifications,aswellasrelatedtop-
ics such as flows and cuts. With wide applications in the fields of communication,
transportation,andproduction,graphconnectivityhasmadetremendousalgorithmic
progressundertheinfluenceoftheoryofcomplexityandalgorithmsinmoderncom-
puter science. New concepts and graph theory algorithms that provide quicker and
moreefficientcomputing,suchasMA(maximumadjacency)orderingofvertices,are
comprehensivelydiscussed.
Covering both basic definitions and advanced topics, this book can be used as a
textbookingraduatecoursesofmathematicalsciences(suchasdiscretemathematics,
combinatorics, and operations research) in addition to being an important reference
bookforallspecialistsworkingindiscretemathematicsanditsapplications.
HiroshiNagamochiisaprofessorattheGraduateSchoolofInformatics,KyotoUniver-
sity.HeisamemberoftheOperationsResearchSocietyofJapanandtheInformation
ProcessingSociety.
ToshihideIbarakiisaprofessorwithKwanseiGakuinUniversityandprofessoremer-
itusofKyotoUniversity.HeisaFellowoftheACM;OperationsResearchSocietyof
Japan;theInstituteofElectronic,InformationandCommunicationEngineers;andthe
InformationProcessingSociety.
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
FOUNDING EDITOR G.-C. ROTA
EditorialBoard
R.Doran,P.Flajolet,M.Ismail,T.-Y.Lam,E.Lutwak
Thetitlesbelow,andearliervolumesintheseries,areavailablefrombooksellersorfrom
CambridgeUniversityPressatwww.cambridge.org.
90 M.Lothaire AlgebraicCombinatoricsonWords
91 A.A.IvanovandS.V.Shpectorov GeometryofSporadicGroupsII
92 P.McMullenandE.Schulte AbstractRegularPolytopes
93 G.Gierzetal. ContinuousLatticesandDomains
94 S.Finch MathematicalConstants
95 Y.Jabri TheMountainPassTheorem
96 G.GasperandM.Rahman BasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.) CategoricalFoundations
98 M.E.H.Ismail ClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.Mora SolvingPolynomialEquationSystemsII
100 E.OlivieriandM.Eula´liaVares LargeDeviationsandMetastability
101 A.Kushner,V.Lychagin,andV.Rubtsov ContactGeometryandNonlinear
DifferentialEquations
102 L. W.Beineke,R.J.Wilson,andP.J.Cameron(eds.) TopicsinAlgebraicGraph
Theory
103 O.Staffans Well-PosedLinearSystems
104 J.M.Lewis,S.Lakshmivarahan,andS.Dhall DynamicDataAssimilation
105 M.Lothaire AppliedCombinatoricsonWords
106 A.Markoe AnalyticTomography
107 P.A.Martin MultipleScattering
108 R.A.Brualdi CombinatorialMatrixClasses
110 M.-J.LaiandL.L.Schumaker SplineFunctionsonTriangulations
111 R.T.Curtis SymmetricGenerationofGroups
112 H.Salzmann,T.Grundho¨fer,H.Ha¨hl,andR.Lo¨wen TheClassicalFields
113 S.PeszatandJ.Zabczyk StochasticPartialDifferentialEquationswithLe´vy
Noise
114 J.Beck CombinatorialGames
116 D.Z.ArovandH.Dym J-ContractiveMatrixValuedFunctionsandRelated
Topics
117 R.Glowinski,J.-L.LionsandJ.He ExactandApproximateControllabilityfor
DistributedParameterSystems
118 A.A.BorovkovandK.A.Borovkov AsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´ GeometryofChemicalGraphs
120 T.Nishiura AbsoluteMeasurableSpaces
121 F.King HilbertTransforms
122 S.Khrushchev OrthogonalPolynomialsandContinuedFractions:FromEuler’s
PointofView
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encyclopedia of mathematics and its applications
Algorithmic Aspects of Graph Connectivity
HIROSHI NAGAMOCHI
KyotoUniversity
TOSHIHIDE IBARAKI
KwanseiGakuinUniversity
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CAMBRIDGEUNIVERSITYPRESS
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Informationonthistitle:www.cambridge.org/9780521878647
(cid:1)C HiroshiNagamochiandToshihideIbaraki2008
Thispublicationisincopyright.Subjecttostatutoryexception
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noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2008
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Nagamochi,Hiroshi,1960–
Algorithmicaspectsofgraphconnectivity/HiroshiNagamochiandToshihideIbaraki.
p. cm.
Includesindex.
ISBN978-0-521-87864-7(hardback)
1.Graphconnectivity. 2.Graphalgorithms. I.Ibaraki,Toshihide. II.Title.
QA166.243.N34 2008
511(cid:2).5–dc22 2008007560
ISBN 978-0-521-87864-7hardback
CambridgeUniversityPresshasnoresponsibilityfor
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Websitesis,orwillremain,accurateorappropriate.
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Contents
Preface pageix
Notation xi
1 Introduction 1
1.1 PreliminariesofGraphTheory 1
1.2 AlgorithmsandComplexities 13
1.3 FlowsandCuts 20
1.4 ComputingConnectivities 34
1.5 RepresentationsofCutStructures 45
1.6 ConnectivitybyTrees 57
1.7 TreeHypergraphs 60
2 MaximumAdjacencyOrderingandForestDecompositions 65
2.1 SpanningSubgraphsPreservingConnectivity 65
2.2 MAOrdering 73
2.3 3-Edge-ConnectedComponents 86
2.4 2-ApproximationAlgorithmsforConnectivity 100
2.5 FastMaximum-FlowAlgorithms 107
2.6 TestingChordality 112
3 MinimumCuts 114
3.1 PendentPairsinMAOrderings 114
3.2 AMinimum-CutAlgorithm 117
3.3 s-Properk-Edge-ConnectedSpanningSubgraphs 119
3.4 AHierarchicalStructureofMAOrderings 123
3.5 MaximumFlowsBetweenaPendentPair 127
3.6 AGeneralizationofPendentPairs 130
3.7 PracticallyEfficientMinimum-CutAlgorithms 131
4 CutEnumeration 137
4.1 EnumeratingAllCuts 137
4.2 EnumeratingSmallCuts 140
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viii Contents
4.3 EnumeratingMinimumCuts 145
4.4 UpperBoundsontheNumberofSmallCuts 149
5 CactusRepresentations 153
5.1 CanonicalFormsofCactusRepresentations 153
5.2 (s,t)-CactusRepresentations 171
5.3 ConstructingCactusRepresentations 180
6 ExtremeVertexSets 191
6.1 ComputingExtremeVertexSetsinGraphs 192
6.2 AlgorithmforDynamicEdgesIncidenttoaSpecifiedVertex 198
6.3 OptimalContractionOrdering 200
6.4 Minimumk-SubpartitionProblem 207
7 EdgeSplitting 217
7.1 Preliminaries 217
7.2 EdgeSplittinginWeightedGraphs 220
7.3 EdgeSplittinginMultigraphs 226
7.4 OtherSplittings 232
7.5 Detachments 237
7.6 ApplicationsofSplittings 240
8 ConnectivityAugmentation 246
8.1 IncreasingEdge-ConnectivitybyOne 247
8.2 StarAugmentation 249
8.3 AugmentingMultigraphs 252
8.4 AugmentingWeightedGraphs 254
8.5 MoreonAugmentation 276
9 SourceLocationProblems 282
9.1 SourceLocationProblemUnderEdge-Connectivity
Requirements 283
9.2 SourceLocationProblemUnderVertex-Connectivity
Requirements 295
10 SubmodularandPosimodularSetFunctions 304
10.1 SetFunctions 304
10.2 MinimizingSubmodularandPosimodularFunctions 306
10.3 ExtremeSubsetsinSubmodularandPosimodularSystems 315
10.4 OptimizationProblemsoverSubmodularandPosimodular
Systems 320
10.5 ExtremePointsofBasePolyhedron 336
10.6 MinimumTransversalinSetSystems 342
Bibliography 357
Index 371
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Preface
Because the concept of a graph was introduced to represent how objects are
connected,itisnotsurprisingthatconnectivityhasbeenacentralnotioningraph
theory since its birth in the 18th century. Various definitions of connectivities
havebeenproposed,forexample,edge-connectivity,vertex-connectivity,andtheir
ramifications.Closelyrelatedtoconnectivityareflowsandcutsingraphs,where
thecutmayberegardedasadualconceptofconnectivityandflows.
A recent general trend in the research of graph theory appears as a shift to
its algorithmic aspects, and improving time and space complexities has been a
strongincentivefordevisingnewalgorithms.Thisisalsotruefortopicsrelatedto
connectivities,flows,andcuts,andmuchimportantprogresshasbeenmade.Such
topicsincludecomputation,enumeration,andrepresentationofallminimumcuts
and small cuts; new algorithms to augment connectivity of a given graph; their
generalization to more abstract mathematical systems; and so forth. In view of
these,itwouldbeatimelyattempttosummarizethoseresultsandpresentthemin
aunifiedsettingsothattheycanbesystematicallyunderstoodandcanbeapplied
tootherrelatedfields.
In these developments, we observe that a simple tool known as maximum
adjacency (MA) ordering has been a profound influence on the computational
complexityofalgorithmsforanumberofproblems.Itisdefinedasfollows.
MA ordering: Given a graph G =(V,E), a total ordering σ =(v ,
1
v2,...,vn) of vertices is an MA ordering if |E(Vi−1,vi)|≥|E(Vi−1,vj)|
holds for all i, j with 2≤i < j ≤n, where V ={v ,v ,...,v } and
i 1 2 i
E(V(cid:2),v)isthesetofedgesfromverticesinV(cid:2) tov.
To our knowledge, MA ordering was first introduced in a paper by R. E. Tarjan
and M.Yannakakis [300],where itwas called theMaximum Cardinality Search
and used to test chordality of graphs, to test acyclicity of hypergraphs, and to
solveotherproblems.WethenrediscoveredMAordering[232],showingthatitis
effectiveforproblemssuchasfindingaforestdecompositionandcomputingthe
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