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Lattices and Polyhedra from Graphs LATTICES AND POLYHEDRA FROM GRAPHS vorgelegt von Diplom-Mathematiker Kolja Knauer aus Oldenburg Von der Fakultat¨ II – Mathematik und Naturwissenschaften der Technischen Universitat¨ Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. rer. nat. Jorg¨ Liesen Berichter: Prof. Dr. rer. nat. Stefan Felsner Prof. Dr. rer. nat. Michael Joswig Tag der wissenschaftlichen Aussprache: 9. November 2010 Berlin 2010 D 83 ACKNOWLEDGEMENTS My work related to this thesis was mainly done in Berlin – me being a part of the research group Diskrete Strukturen. This was really nice mainly because of the members of this group and the communicative, relaxed, and productive atmosphere in the group. Thank you a lot: Daniel Heldt, Stefan Felsner, Andrea Hoffkamp, Mareike Massow, Torsten Ueckerdt, Florian Zickfeld. I am thankful for Stefan’s helpful advice on how to do, write and talk math. I really enjoyed working with him as my advisor and also as my coauthor. I also want to thank my Mexican coauthors Ricardo Gomez,´ Juancho Montellano-Ballesteros and Dino Strausz for the nice time working and discussing at UNAM in Mexico. My stays in Mexico would not have been possible without the help of Isidoro Gitler. Ulrich Knauer, Torsten, Gunter¨ Rote, Eric´ Fusy made helpful comments and fruitful discussions. Ulrich and Torsten moreover helped me a big deal by proofreading parts of the thesis. Another big helper who made this work possible was SOLAC with its 18bar of pure pressure and many cups of coffee. I should also mention that this work would not have been possible without the funding by the DFG Research Training Group “Methods for Discrete Structures”. Finally, I want to express my gratitude again to Stefan and also to Michael Joswig for reading this thesis. I hope you enjoy some of it, but I know this is quite some work and so I stumbled over something that Douglas Adams [1] said and which seemed adequate to me: “Would it save you a lot of time if I just gave up and went mad now?” Thanks a lot, Kolja Contents What is this thesis about? 1 1 Lattices 5 1.1 PreliminariesforPosetsandLattices . ...... 9 1.2 GeneralizingBirkhoff’sTheorem. .... 13 1.2.1 Applications ............................. 24 1.2.2 Duality ................................ 26 1.3 Hasse Diagrams of Upper Locally Distributive Lattices . .......... 28 1.4 TheLatticeofTensions. 38 1.4.1 Applications ............................. 44 1.4.2 The latticeof c-orientations(Propp[92]) . 44 1.4.3 The lattice of flows in planar graphs (Khuller, Naor and Klein[66]) 46 1.4.4 Planar orientations with prescribed outdegree (Felsner [39], Ossona deMendez[88]) ........................... 48 1.5 Chip-Firing Games, Vector Addition Languages, and Upper Locally Dis- tributiveLattices ............................... 50 1.6 Conclusions.................................. 61 2 Polyhedra 65 2.1 PolyhedraandPosetProperties . ... 68 2.2 AffineSpace.................................. 71 2.3 UpperLocallyDistributivePolyhedra . ..... 75 i ii 2.3.1 FeasiblePolytopesofAntimatroids . .. 80 2.4 DistributivePolyhedra . 84 2.4.1 TowardsaCombinatorialModel . 85 2.4.2 TensionsandAlcovedPolytopes . 87 2.4.3 GeneralParameters. 90 2.4.4 PlanarGeneralizedFlow . 96 2.5 Conclusions.................................. 98 3 CocircuitGraphsofUniformOrientedMatroids 103 3.1 PropertiesofCocircuitGraphs . 105 3.2 TheAlgorithm ................................ 108 3.3 Antipodality.................................. 110 Bibliography 112 Notation Index ................................... 121 Index ........................................ 123 What is this thesis about? The title “Lattices and Polyhedra from Graphs” of this thesis is general though describes quite well the aim of this thesis. Among the most important objects of this work are dis- tributive lattices and upper locally distributive lattices. While distributive lattices certainly are one of the most studied lattice classes, also upper locally distributive lattices enjoy fre- quent reappearance in combinatorial order theory under many different names. Upper locally distributive lattices correspond to antimatroids and abstract convex geometries – objects of major importance in combinatorics. Besides results of a purely lattice or order theoretic kind we present new characterizations of (upper locally) distributive lattices in terms of antichain-covers of posets, arc-colorings of digraphs, point sets in Nd, vector addition languages, chip-firing games, and vertex and (integer) point sets of polyhedra. We exhibit links to a wide range of graph theoretical, combinatorial, and geometrical objects. With respect to the latter we study and characterize polyhedra which seen as subposets of the componentwise ordering of Euclidean space form (upper locally) distributive lattices. Distributive lattice structures have been constructed on many sets of combinatorial objects, such as lozenge tilings, planar bipartite perfect matchings, pla- nar orientations with prescribed outdegree, domino tilings, planar circu- lar flow, orientations with prescribed number of backward arcs on cycles and several more. A common feature of all of them is that the Hasse di- agram of the distributive lattice may be constructed applying local trans- formations to the objects. These local transformations lead to a natural arc-coloring of the diagram. For an example see the distributive lattice on the domino-tilings of a rectangular region on the side. The local transfor- mation consists in flipping two tiles, which share a long side. In this work we present the first unifying generalization of all such instances of graph- related distributive lattices. We obtain a distributive lattice structure on the tensions of a digraph. In order to provide a flavor of what we refer to as “unifying generalization”, we show two consecutive steps of generalizing the domino tilings of a plane region, see the figure. 1 2 The left-most part of the figure shows two domino tilings, which can be transformed into each other by a single flip of two neighboring tiles. In the middle of the figure we show how planar bipartite perfect matchings model domino tilings. The local transformation now cor- responds to switching the matching on an alter- nating facial cycle. More generally, the right-most part of the figure shows how to interpet the pre- ceeding objects as orientations of prescribed out- degree of a bipartite planar graph. Every yellow vertex has outdegree 1 and every blue vertex has outdegree (deg − 1). Reversing the orientation on directed facial cycles yields a distributive lattice structure on the set of orientations with these outdegree constraints. A particular interest of this work lies in embedding lattices into Euclidean space. The motivation is to combine geometrical and order-theoretical methods and perspectives. We investigate polyhedra, which seen as subposets of the componentwise ordering of Euclidean space form upper locally distributive or distributive lattices. In both cases we obtain full characterizations of these classes of polyhedra in terms of their description as intersection of bounded halfspaces. In particular we obtain a polyhedral structure on known discrete distributive lattices on combinatorial objects as those mentioned above as integer points of distributive polytopes. A classical polytope which was defined in the spirit of combining discrete geometry and order-theory appears as a special case of our considerations, and thus might provide an idea of what kind of objects we will study: Given a poset P, Stanley’s order polytope PP may be defined as the convex hull of the characteristic vectors of the ideals of a poset P. z y z y x x Figure 1: A poset P with an ideal on the left with its order polytope PP and the vertex the corresponding to the ideal, on the right. Our characterization of upper locally distributive polyhedra opens connections to the the- ory of feasible polytopes of antimatroids. In the setting of distributive polyhedra we find graph objects that might be considered as the most general ones, which form a distributive lattice and carry a polyhedral structure. The connection to polytope theory links distributive lattices to generalized flows on digraphs. Thus, there is a link to important objects of com- INTRODUCTION 3 binatorial optimization. Moreover we exhibit new contributions to the theory of bicircular oriented matroids. Large parts of the thesis are based on publications between 2008 and 2010 [40, 43, 41, 42, 54, 69]. In the following we give a rough overview over each single chapter. For more detailed introductions we refer to the first pages of the individual chapters. Chapter 1: Lattices The first chapter of the thesis is about lattices. It is based on papers [41, 42, 69] and includes joint work with Stefan Felsner. After giving a more detailed introduction into lattice theory and the chapter itself, we present some basic notation and vocabulary in Section 1.1. The main result of Section 1.2 is a new representation result for general finite lattices. We provide a one-to-one correspondence between finite lattices and antichain-covered posets. As an application we strengthen a characterization of upper locally distributive lattices in terms of antichain-partitioned posets due to Nourine. The “smallest” special case of our the- orem is the Fundamental Theorem of Finite Distributive Lattices alias Birkhoff’s Theorem. Section 1.3 proves three classes of combinatorial objects to be equivalent. We show that
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