Cover Graphs and Order Dimension vorgelegt von M. Sc.{Mathematiker Veit Wiechert aus L¨ubz Von der Fakult¨atII Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften { Dr. rer. nat. { genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. J¨orgLiesen Berichter: Prof. Dr. Stefan Felsner Prof. Dr. Tibor Szab´o Dr. Piotr Micek Tag der wissenschaftlichen Aussprache: 23. Februar 2017 Berlin 2017 Acknowledgments This dissertation is the result of three years of research within the Discrete Mathe- matics Group at TU Berlin. I like to express my greatest gratitude to my advisor Stefan Felsner for introducing me to the world of order dimension and for every kind of support. I enjoyed big freedom under his supervision and was always free to choose my favorite research problem. Stefan is also the reason why his working group is known for its relaxed and productive atmosphere. I also like to thank my (former) colleagues from this group for the fantastic time here: Andrea Hoffkamp, Kolja Knauer, Daniel Heldt, Torsten Ueckerdt, Irina Mustat¸ˇa,Nieke Aerts, Udo Hoffmann, Linda Kleist, and Torsten M¨utze. Special thanks go to my office mate Linda for the many funny moments we shared over the last six months. Another important person of my academic career is Tom Trotter. He is pushing and contributing to dimension theory for many decades now. Still, he seems to be as enthusiastic about posets and their dimension as he was from the beginning on. I am grateful to him for sharing this enthusiasm in the numerous discussions we had. I also would like to thank him for giving me the opportunity to speak at the minisymposium he organized around the SIAM Conference on Discrete Mathematics in Halifax in 2012. This event plays a special role in my career as it strongly influenced my research interests. I would like to thank my colleagues and friends Gwena¨elJoret and Piotr Micek for many exciting research sessions. Especially the week we spent together in Brussels in 2015 was a wonderful experience. We worked an uncountable number of hours on two of our papers; needless to say that I learned a lot about writing mathematics from Gwen and Piotr. While his 2-years stay at our working group, Piotr introduced me to many interesting math problems. I am grateful to him for our regular morning sessions, which led to several joint publications. Last and not least, I very much want to thank Johanna and our daughter Helena for the beautiful time besides the academic world. Thank you all, Veit Contents Introduction and Overview 1 1 Preliminaries 7 1.1 Posets, Cover Graphs, and Dimension .................. 7 1.2 Graph Minors ............................... 12 2 Essential Tools for Dimension 15 2.1 Min-Max Reduction ........................... 16 2.2 Unfolding Posets and Cores ....................... 17 2.3 Zig-Zag Paths ............................... 22 3 Planar Posets 26 3.1 Planar Posets with Support ....................... 28 3.2 A Linear Upper Bound on the Dimension . 45 3.3 New Lower Bound Constructions .................... 53 4 Tree-width, Path-width, and Tree-depth 55 4.1 Cover Graphs with Bounded Tree-depth . 57 4.2 Cover Graphs with Tree- or Path-width 2 . 59 4.3 Cover Graphs with Path-width at most p . 66 4.4 Lower Bound Constructions ....................... 70 5 Cover Graphs with Excluded Topological Minor 76 5.1 First Case: Posets of Bounded Height . 77 5.2 Second Case: (k + k)-free Posets .................... 85 6 Sparsity and Dimension 90 6.1 Classes with Bounded Expansion and Nowhere Dense Classes .......................... 92 6.2 Weak Coloring Numbers and Dimension . 95 6.3 Application: Boxicity of Graphs . 103 Conclusions and Open Problems 106 Bibliography 111 Index 117 Introduction and Overview The goal of this dissertation is to study the various connections between the dimen- sion of posets and graph theoretic properties of their cover graphs. The dimension of a poset P is the least number of linear extensions of P whose intersection gives rise to P . This parameter was introduced in 1941 by Dushnik and Miller [17] and turned out to be one of the most important concepts when studying the combinatorics of posets. Computing the dimension is a classical computationally hard problem. The decision problem whether the dimension of a poset is at most k, for some fixed k, was posed as one of the twelve problems on Garey's and Johnson's famous list of problems with unknown complexity in 1979. While it is decidable in polynomial time whether a poset has dimension at most 2, the problem becomes NP-complete once k is at least 3, as shown by Yannakakis in 1982 [77]. The problem even remains NP-complete if only posets of height 2 are under consideration (Felsner et al. [25]). Giving good approximations is hard as well: Chalermsook et al. [10] argued that it is unlikely that there exists a polynomial-time algorithm that approximates the 1 dimension within a factor of (n − ), where n denotes the number of elements in the poset. O Despite all these hardness results, we want to understand this parameter and try to find sufficient conditions and witnesses of large-dimensional posets. On the other hand, we want to determine properties that are necessarily satisfied by them. A number of such properties are known; for instance, Hiraguchi showed that every n-dimensional poset (n > 3) has to contain at least 2n elements [36]. Moreover, when n is large, then some of the poset elements must be involved in many compa- rabilities (R¨odl,Trotter [67]; F¨uredi,Kahn [30]). Another such property is a simple consequence of a classic theorem of Dilworth [15]: the width of a poset, which is the maximum size of a contained antichain, is at least as large as the dimension of the poset. This fact can be rephrased as follows: Large-dimensional posets have to be wide. The situation is different for the height of a poset, which is maximum size of a chain in the poset. The so-called standard example of order n has dimension n, while its height is only 2; Figure 1 (left) illustrates such an example of order 4. Therefore, large-dimensional posets need not be tall. In the following we discuss structural graph properties that are necessarily sat- isfied in cover graphs of posets with large dimension. Cover Graphs. Posets are usually visualized by their Hasse diagrams, in which every cover relation is represented by a y-monotone curve. The abstract undirected graph behind the Hasse diagram is the cover graph of the poset. Another graph that we associate with a poset is its comparability graph, where two elements are joined by an edge if they are comparable. Those two graphs behave differently in 1 b4 b1 b2 b3 b4 b3 b2 b1 a1 a2 a3 a4 a1 ai < bj iff i = j a 6 2 a3 a4 Figure 1: Standard example (left), and Kelly's example containing it as an induced subposet (right). many aspects: While width, height, and also dimension are invariant on posets with the same comparability graph, this is not true for cover graphs. The dimension of posets with the same cover graph may even differ by an arbitrary number. However, there are structural properties of graphs G that yield valuable information about the dimension of (all) posets whose cover graphs are isomorphic to G. Another fun- damental difference between these two graph concepts is given by the computational complexity of their corresponding decision problems: It is well known that deciding whether a graph is a comparability graph can be done in polynomial time. On the other hand, testing whether a graph is a cover graph is an NP-complete problem as shown by Neˇsetˇriland R¨odl[58], and Brightwell [9]. Why studying cover graphs? First of all, these are the ones that we use to visual- ize posets and hence they encode the topology of posets. Another important reason to consider cover graphs is that they are usually sparser than comparability graphs, and this sparsity may admit the existence of an underlying topological structure that can be used to give good estimations on the dimension. An example where such a structure can be exploited is given by the class of planar posets. A poset is planar if its Hasse diagram can be drawn without any edge crossings. In 1972, Baker, Fishburn, and Roberts proved that planar posets with a single minimal element and a single maximal element have dimension at most 2 [2]. The first result in this context that is using only the structural properties of cover graphs is due to Moore and Trotter; they show that posets with trees as cover graphs have dimension at most 3 [75]. Dimension versus Height. These early results led to the question whether the dimension of planar posets is bounded by a constant. It was answered in the neg- ative by Kelly in 1981, who constructed an explicit family of planar posets that contain arbitrarily large standard examples and hence have unbounded dimension; see Figure 1 (right). After Kelly's discovery, researchers in the field lost interest in studying connections between the topological structure of cover graphs and di- mension for decades. Probably, the main reason here was that Kelly's result broke a possible analogy between order dimension and chromatic number: While planar graphs have chromatic number at most 4, there is no such constant bound on the dimension of planar posets. This interest got renewed only after Felsner, Li, and Trotter [24] discovered in 2010 that the dimension of height-2 posets with planar 2 cover graphs is at most 4.