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Interval Orders Interval Orders Combinatorial Structure and Algorithms im Februar von Dipl Math Stefan Felsner am Fachb ereich Mathematik der Technischen Universitat Berlin vorgelegte Dissertation D Preface This thesis is based on my research on partially ordered sets and sp ecially inter val orders that b egan when I came to Berlin in Professor RH Mohring intro duced me to the eld I like to thank him for stimulations and guidance over the years Beside the intro duction the thesis combines ve chapters that have in common the central role played by interval orders On the other hand they are only lo osely connected and so I decided to make the chapters selfcontained To emphasize the indep endency of the chapters references are given at the end of each one Articles and b o oks that are of signicance to the general theme and have b een consulted without b eing cited directly are collected in the references of the intro duction An outline of the contents of the thesis can b e found in the preview at the b eginning of the intro duction All further chapters are op ened by a section called Intro duction and Overview That sp ecial section may serve as an extended abstract for the contents of the chapter it also gives the relationship to the existing literature I am indebted to many p eople for encouragement and discussions Sp ecial thanks go to Tom Trotter Lorenz Wernisch and the memb ers of our group dis crete algorithmical mathematics which provided an ageeable and creative atmo sphere Stefan Felsner Berlin February Contents Intro duction Preview Partially Ordered Sets Interval Orders References for Chapter The Jump Numb er of Interval Orders Intro duction and Overview Greedy Linear Extensions and Starting Elements Bounds for the Jump Numb er of Interval Orders An Approximation Algorithm Defect Optimal Interval Orders The NP Completeness Pro of References for Chapter The Dimension of Interval Orders Intro duction and Overview Some Logarithmic Bounds Marking Functions and Bounds The StepGraph and More Bounds Op en Problems The Doubly Logarithmic Bound Canonical Interval Orders General Interval Orders Colorings of Digraphs and ShiftGraphs Colorings of Directed Graphs Asymptotics CONTENTS A Connection with Lattices of Antichains References for Chapter Coloring Interval Order Diagrams Intro duction and Overview A Construction of Long Sequences The Structure of Very Long Sequences Long Cycles b etween Consecutive Levels in B n References for Chapter Interval Dimension and Dimension Intro duction and Overview Reducing Interval Dimension to Dimension Consequences Crowns and the Standard Examples B and the Split Op eration The Interval Dimension of Sub divisions Comparability Invariance of Interval Dimension Remarks on Computational Complexity References for Chapter Tolerance Graphs Intro duction and Overview Tolerance Graphs and Interval Dimension Some Examples Cotrees and More Examples References for Chapter Chapter Intro duction Preview Here we only give a very brief survey of the contents of this thesis For more detailed information on the sub ject of a single chapter we refer to intro ductory sections called Intro duction and Overview there we also relate our own results to the existing literature A family I of intervals on the real line may b e used to dene a partial x xX ordering on X We put x y if a I and b I implies a b A partial order x y I X thus obtained is called an interval order In this chapter we intro duce basic terminology and facts ab out partial orders linear extensions and interval orders that are used throughout all parts The next three chapters then deal with problems for interval orders that are known to b e NP complete for general partial orders In the next chapter we consider the jump number problem That is we are lo oking for a linear extension of I which has a minimal numb er of adjacent pairs that are incomparable in I We intro duce parameters and of an interval order and prove a close relationship with the jump numb er The jump numb er of an interval order is at least max A linear extension with at most jumps can always b e found This is an approximation ratio of There is a p olynomial algorithm which decides whether the jump numb er is exactly It is NP complete to decide whether the jump numb er is exactly Chapter deals with the dimension of interval orders The dimension of a partial CHAPTER INTRODUCTION order is the minimal dimension of an euklidean space that admits an emb edding of the order We rst use the concept of marking intervals to obtain easy pro ofs for some logarithmic b ounds Afterwards we intro duce the stepgraph of an interval order A partition of the arcs of the stepgraph into semitransitive classes leads to a realizer This observation is used to give more b ounds particularly a doubly logarithmic b ound Motivated by the pro of of the lower b ound for the dimension of interval orders we nally study colorings and arccolorings of digraphs and stepgraphs In Chapter we directly deal with colorings namely with the chromatic num b er of the diagram of an interval order It is shown that log n colors always suce for diagrams of hight n On the other hand there are interval orders of this hight such that the diagram requires log n colors The construction of go o d colorings relies on the existence of certain sequences of sets of colors sequences An upp er b ound for the length of sequences is given and we show that sequences of this length corresp ond to level accurate Hamiltonian paths in the Bo olean lattice In Bo olean lattices of order we could construct such paths Chapter deals with the interplay of interval dimension and dimension We dene a transformation P Q b etween partial orders such that the dimension of Q and the interval dimension of P agree We provide two interpretations of this transformation a combinatorial one and a geometrical one These two in terpretations are used for several consequences We relate the interval dimension of sub divisions of an order P to the dimension of P We give a new pro of for the comparability invariance of interval dimension Finally in the last chapter we deal with tolerance graphs The complement of a b ounded tolerance graph has an orientation as partial order of interval di mension This order admits a square representation while general orders of interval dimension require nonsquare rectangels in their representation This observation is the starting p oint for our invertigations ab out the relations b e tween several classes of graphs Some of our results are If the complement of a tolerance graph admits an orientation as partial order then this order has inter val dimension We give an example of an alternatingly orientable graph that is the complement of an order of interval dimension but is not a tolerance graph We also characterize the tolerance graphs among the complements of trees PARTIALLY ORDERED SETS Partially Ordered Sets In this and the next section we intro duce necessary terminology and basic results ab out partially ordered sets linear extensions and interval orders A strict partial ordering R on a nite set X is a binary relation on X such that x x is not in R for any x X if x y and y z are in R then x z is in R Condition is the irreexivity of R and Condition says that R is transitive Note that as a consequence of the two conditions we obtain that R is antisym metric ie if x y R and x y then y x is not in R In some situations it is appropriate to use the idea of a reexive partial ordering This is a relation R which is reexive ie x x is in R for each x X antisymmetric and transitive The dierence b etween a strict and a reexive ordering is normally clear from the context and the word ordering may refer to either kind of ordering A set X together with a partial ordering R on X is called a partial ly ordered set and denoted by X R We often abbreviate partially ordered set as either ordered set order or poset The commonly used notation for a partially ordered set P is P X in the case of a strict ordering and P X if the ordering is reexive We then write x y instead of x y and y x to mean x y There is a further abuse of notation which should b e mentioned If P X is a p oset we sometimes write x P or x y P instead of x X and x y The comparability graph of a p oset P X denoted CompP is a graph on X its edges are the comparabilities of P ie fx yg is an edge if either x y or y x A binary relation R on a set X can b e represented graphically by a directed graph digraph for short We represent the elements of X as p oints and use arrows arcs to represent the ordered pairs in R When the binary
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