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Cycle Bases of Graphs and Spanning Trees with Many Leaves

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Cycle Bases of Graphs and Spanning Trees with Many Leaves Cycle Bases of Graphs and Spanning Trees with Many Leaves Complexity Results on Planar and Regular Graphs Von der Fakultät für Mathematik, Naturwissenschaften und Informatik der Brandenburgischen Technischen Universität Cottbus - Senftenberg zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation vorgelegt von Diplom-Mathematiker Alexander Reich geboren am 29.10.1982 in Lübben Gutachter: Prof. Dr. Ekkehard G. Köhler Gutachter: Prof. Dr. Rolf H. Möhring Gutachter: Prof. Dr. Romeo Rizzi Tag der mündlichen Prüfung: 15. April 2014 Abstract A cycle basis of a graph is a basis of its cycle space, the vector space which is spanned by the cycles of the graph. Practical applications for cycle bases are for example the optimization of periodic timetables, electrical engineering, and chemistry. Often, cycle bases belong to the input of algorithms concerning these fields. In these cases, the running time of the algorithm can depend on the size of the given cycle basis. In this thesis, we study the complexity of finding minimum cycle bases of several types on different graph classes. As a main result, we show that the problem of minimizing strictly fundamental cycle bases on planar graphs is -complete. We also give a similarly structured proof for problem of finding a maximumN P leaf spanning tree on the very restricted class of cubic planar graphs. Additionally, we show that this problem is -complete on k-regular graphs for odd k greater than 3. Furthermore, we classify typesAPX of robust cycle bases and study their relationship to fundamental cycle bases. Zusammenfassung Eine Kreisbasis eines Graphen ist eine Basis des Zyklenraumes des Graphen, also des Vektorraumes, der von den Kreisen des Graphen aufgespannt wird. Praktische Anwen- dungsgebiete von Kreisbasen finden sich zum Beispiel bei der Optimierung von Taktfahrplä- nen, in der Elektrotechnik und der Chemie. Oft gehören Kreisbasen zur Eingabe von Algo- rithmen aus diesen Gebieten. In diesen Fällen kann die Laufzeit der Algorithmen von der Größe der eingegebenen Kreisbasis abhängen. In dieser Arbeit untersuchen wir die Kom- plexität, minimale Kreisbasen unterschiedlicher Typen auf verschiedenen Graphenklassen zu finden. Als ein Hauptergebnis zeigen wir, dass das Problem minimale streng funda- mentale Kreisbasen von planaren Graphen zu finden -vollständig ist. Wir geben einen ähnlich strukturierten Beweis für das Problem, auf derN sehr P eingeschränkten Graphenklasse der kubischen planaren Graphen einen spannenden Baum mit der maximalen Anzahl an Blättern zu finden. Zusätzlich zeigen wir, dass dieses Problem auf k-regulären Graphen für ungerade k größer als 3 -vollständig ist. Ein weiteres Thema der Arbeit ist die Klassifizierung von TypenAPX robuster Kreisbasen und ihre Beziehung zu fundamentalen Kreisbasen. Acknowledgements Since my first contact with cycle bases of graphs as a student in 2006 I met a number of people who raised my interest in this topic further and further. At this place, I want to take this opportunity of thanking them for all of their valuable advices and inspirations in a chronological order. First of all, I wish to thank Ekkehard Köhler for organizing my internship at the Technical University of Berlin. The same thanks must also go to Christian Liebchen and Gregor Wünsch for their intensive supervision during that time. Also my diploma thesis and my first publication arose from this internship. It had been a pleasure for me to participate in the Workshop on Cycle and Cut Bases which took place at the Eberhard Karls University, Tübingen in May 2008. In these days, I became truly inspired in many problems concerning cycle bases which had been previously unknown to me. Thank you to all organizers of this workshop, especially to Katharina Zweig, and again to Christian Liebchen, who had drawn my attention to this workshop. Thank you also to the organizers of the 9th SEG Workshop über Kombinatorik, Graphen- theorie und Algorithmen which was held at the Chemnitz University of Technology in June 2011. In August 2012, FRICO took place at the Zuse Institute Berlin. At this workshop, I could present some results of this thesis, especially on spanning trees with many leaves. Thanks to the organizers of FRICO 2012. Now, as the thesis is accomplished, I am especially grateful to my supervisor Ekkehard Köhler. Thank you for your support, encouragement, motivation, and all the helpful suggestions which I received from you. I also want to thank my assessors Rolf H. Möhring and Romeo Rizzi. I do not want to forget to thank all members of staff on the third floor in the Main Building of BTU Cottbus for the supply of an efficient, motivating, and simply optimum working environment. In particular, I want to thank Martin Strehler and my former room- mate Harry Schülzke. Furthermore, I want to thank all proofreaders of the manuscript, in particular Martin and Randolf, as well as Uwe for his technical support. A really special thank you is also due to Diana Hübner. Finally, I want to express my gratitude to my family, my friends in Sellendorf, and to my girlfriend Jana Paulick. Cottbus, October 2013 Alexander Reich Contents Introduction 11 1 Preliminaries 15 1.1 GraphTheory.................................. 15 1.2 CyclesandCycleSpaces ............................ 18 1.3 Complexity and Approximation . .. 22 2 Spanning Trees with Many Leaves 29 2.1 Introduction................................... 29 2.2 Applications................................... 31 2.3 -completeness of the MLST onPlanarCubicGraphs . 33 N P 2.3.1 Connectionofthe3-Sets . 34 2.3.2 TheProblems.............................. 35 2.3.3 TheTransformation .......................... 35 2.3.4 Restriction to Biconnected Graphs . .. 39 2.4 -Completeness of the MLST onRegularGraphs. 41 APX 2.4.1 TheProblems.............................. 41 2.4.2 TheTransformation .......................... 42 2.4.3 Extension to Graphs with Arbitrary Odd Regularity . ...... 45 2.5 Algorithms for Selected Graph Classes . ..... 48 2.6 ConclusionsandOutlooks . 49 7 8 CONTENTS 3 Strictly Fundamental Cycle Bases 51 3.1 Introduction................................... 52 3.2 Applications................................... 53 3.3 Basic Definitions and Properties on SFCBs . ..... 55 3.4 SFCBsonPlanarGraphs............................ 59 3.4.1 DefinitionsforPlanarGraphs . 59 3.4.2 SFCBsonWeightedPlanarGraphs . 61 3.4.3 Relationship to the OCST Problem................... 63 3.4.4 SFCBsonNon-MetricPlanarGraphs . 65 3.5 -completeness of the MSFCB inPlanarGraphs. 68 N P 3.5.1 TheProblems.............................. 68 3.5.2 TheTransformation .......................... 69 3.6 OuterplanarGraphs .............................. 81 3.6.1 Introduction............................... 81 3.6.2 Definition and Properties on Outerplanar Graphs . ..... 82 3.6.3 MinorMonotonicity .......................... 84 3.7 CycleRootGraphs ............................... 89 3.8 Conclusions ................................... 92 4 Classification of Robust Cycle Bases 93 4.1 Introduction................................... 93 4.2 Applications................................... 94 4.3 ClassesofRobustCycleBases . .. 96 4.4 Examples of Robust Cycle Bases . 97 4.5 Relationship with Fundamental Bases . ..... 104 4.6 Conclusions ................................... 112 CONTENTS 9 5 Further Classes of Cycle Bases 113 5.1 p-Bases...................................... 114 5.1.1 Definition of p-Bases for Directed and Undirected Graphs . 114 5.1.2 p-Bases for Large p ........................... 115 5.1.3 p-Bases for Small p ........................... 118 5.1.4 Conclusions ............................... 120 5.2 Totally Unimodular Cycle Bases . .. 120 5.2.1 Introduction............................... 120 5.2.2 Basic Definitions and Properties on TUM Cycle Bases . .... 121 5.2.3 TUM Bases vs. Weakly Fundamental Cycle Bases . 124 5.2.4 Conclusions ............................... 132 5.3 IntegralCycleBases .............................. 133 5.3.1 Introduction............................... 133 5.3.2 The PESP for Modeling Periodic Timetables . 134 5.3.3 Definition ................................ 136 5.3.4 An Integral Cycle Basis Without Simple Cycles . .... 137 5.3.5 TUM Bases and the Exchange Property . 138 5.3.6 Conclusions ............................... 140 Bibliography 143 Index 153 List of Problems 157 Introduction Graph drawing. Chemistry. Electrical engineering. Category theory. Periodic timetabling. What do these areas have in common? How does category theory matches up with this list? Beyond others, these fields are applications for different classes of cycle bases, the main topic of this thesis. From an etymological point of view, the word cycle is derived from the Late Latin cyclus respectively from the Greek word κύκλος ([96]) and means “any complete round or series of occurrences that repeats or is repeated” ([35]). Besides cycles, there are three further terms used in this thesis to describe mathematical objects which are round in a sense. In our meanings, circulation is essentially the same as a cycle, while a circuit comes along with a simpler structure. Finally, by a circle we mean what each elementary-school pupil ought to know, namely the set of all points in a plane that have the same distance to a given point. Anyway, circles will not play a major role. Rather, we will deal with circuits and cycles in graphs. The set of all cycles in a graph forms a vector space. And this vector space, which is called the cycle space in this case, has a basis—the
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