Interval Orders

Combinatorial Structure and Algorithms

Februar

von

Dipl Math Stefan Felsner

am Fachb ereich Mathematik

der Technischen Universitat Berlin

vorgelegte Dissertation

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

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

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 relation

is a partial ordering the graphical representation can b e simplied Since the

relation is understo o d to b e transitive we can omit arrows b etween p oints that are

connected by a sequence of arrows When the graphical representation is oriented

such that all arrowheads p oint upwards we can even omit the arrowheads as the

example in Figure shows Such a graphical representation of a p oset in which

all arrowheads are understo o d to p oint upwards is also known as the diagram

CHAPTER INTRODUCTION

or Hasse diagram of the p oset

p oset diagram

h h g h h h g h

I AK

h h h h c d f h h h h c f

e d e

HY H

H H A

H H

H H A

H H

H H A

H H

h h H H h h A

a b a b

Figure A p oset and its diagram

Let P X b e a partially ordered set A subset of X is called a chain if

every two elements in the subset are comparable Note that if fx x x g

is a chain in P then there is some rearrangement of indices such that x

x x We refer to the numb er of elements in a chain as the length

i i

of the chain The height of an element x P denoted by height x is the

length of the longest chain x x x ie of the longest chain ending in

x The height of a p oset P denoted height P is the length of a longest chain

in P A subset of X is called an antichain in the p oset P X if no two

distinct elements in the subset are related A pair x y of unrelated elements is

also called an incomparable pair and denoted by xjjy The width of a partially

ordered set P X denoted width P is the maximal size of an antichain in

P For example in the p oset P of Figure fa c gg fb d gg and fa dg are

chains and fc d e fg fa fg and fb eg are antichains the height of c is while

heightP and widthP

Let P X b e a partially ordered set An element x in P is called a

minimal element if there is no y with y x the set of all minimal elements

of P is denoted by MinP An element x in P is called a maximal element if

there is no y with y x the set of all maximal elements is denoted by MaxP

If P is the p oset of Figure then MinP fa b eg and MaxP fg hg An

element x is said to cover another element y denoted x y if x y and there

is no element z with x z y Note that the covering pairs of an order P are

exactly the edges in the diagram of P

As an illustration of the concepts of chains and antichains in partially ordered

PARTIALLY ORDERED SETS

sets we present two theorems that show a close relationship b etween them

Theorem Let P X be a partial ly ordered set The height of P

equals the minimum number of antichains required to cover the elements

of P

Pro of Let C b e a chain of length height P Since an antichain may contain at

most one element of C a covering of P by antichains requires at least heightP

antichains

For the converse we use induction on the height of P Let height P n

and M MinP Clearly M is a nonempty antichain Consider now the p oset

P X n M and note that heightP n By induction P can b e covered

by n antichains Thus P can b e covered by n antichains 2

The dual of this theorem is given next It is known as Dilworths theorem

Theorem Dilworth Let P X be a partial ly ordered set The

width of P equals the minimum number of chains needed to cover the

elements of P

Pro of Let A b e an antichain of size width P Since any chain may contain at

most one element of A a covering of P by chains requires at least widthP chains

For the converse we use induction on jXj the numb er of elements in P Let C

b e a maximal chain of P If C nontrivially intersects every maximum antichain

then we remove C and use induction Otherwise there is a maximum antichain

U D

A disjoint from C Set A fx X j x a for some a Ag and A fx

X j x a for some a Ag Note that C A would imply by maximality of

D U

C that the minimal element of C is in A Similarly C A Hence jA j jXj

D U D

and jA j jXj By induction A and A b oth can b e covered with jAj chains

which have the elements of A as maximal resp ectively minimal elements At

U D

these p oints the chains of A and A can b e put together Since A is maximal

U D

A A X and we obtain a covering of P by jAj chains 2

The probably most imp ortant family of partial orders are the Bo olean lat

tices The Boolean lattice B is the set of all subsets of f ng ordered by

inclusion ie A B i A B As a nice application of chain coverings we will

now derive another classical theorem of p oset theory It is known as Sp erners theorem

CHAPTER INTRODUCTION

Theorem Sp erner Let A be an antichain of subsets of an nset

Then

jAj

e d

Pro of We use induction on n For n the claim is trivially true

If C S S S is a chain of length in B Then

C S S S S fng and C S fng S fng

S fng are chains in B Now let fC C C g b e a minimal family of

chains covering B By induction t Note that every chain C has

d e

m l

element sets to contain a set from the antichain of

Now consider the collection C consisting of the chains C C if the length of

i i

C is at least together with the chains C if the length of C is one It can b e

i i

seen that C is a chain covering of B Moreover every chain in C contains a set

m l

Therefore C consists of chains and and every antichain in B of size

d e

has at most this size 2

A p oset P X which is a chain is called a linear order Let L X

b e a linear order we then write L x x x as an abbreviation for x

x x An extension Q of a p oset P X is a p oset on the

L L P

same elements with x y whenever x y Of sp ecial imp ortance are those

Q P

extensions of P which are linear orders they are called linear extensions

A linear extension L x x x of P X induces a partition

L C C C of P into chains The chains in this partition are maxi

mal segments of L such that the elements in the segment are pairwise comparable

in P consequently maxC min C for each i The problem of minimizing the

i i

numb er of chains in a partition induced by a linear extension is the jump number

problem In the context of this problem it is useful to view linear extensions as

the result of an algorithmic pro cess A generic algorithm for linear extensions is

Algorithm

Using appropriate sp ecications of the subroutine cho ose we may obtain

every linear extension of P as output of the algorithm With the pro of of the

next lemma we give an application of this freedom of sp ecication

Lemma Let P X be a poset and xjjy Then there is a linear

extension L of P which takes x before y ie x y L

PARTIALLY ORDERED SETS

Algorithm

Linear Extension

L the empty list

for i to n do

cho ose x MinP

L L x

P P n fx g

output L

Pro of Sp ecify the choice in the following way Cho ose any element distinct

from y as long as MinP fyg Using this rule y is chosen exactly when

P fz X j y zg Since x is not in this set we will nd x somewhere b efore y

in L 2

Note that the intersection of two partial orderings is again irreexive and

transitive that is a partial ordering With this in mind we can state another

classical theorem

Theorem Dushnik and Miller

Every poset P is the intersection of its linear extensions

Pro of This is an immediate consequence of Lemma 2

A family of linear extensions of P such that P is their intersection is called a

realizer of P Note that for each incomparable pair xjjy in P there must b e two

linear extensions L L in a realizer with x y and y x we then say the

L L

pair x y is realized by L L The size of the smallest realizer of P X is

called the dimension of P and abbreviated dimP Some authors prefer the more

precise names order dimension or DushnikMil ler dimension Nowadays

dimension theory is a strong branch in the theory of partially ordered sets This

is do cumented by the recent b o ok of Trotter Tr which gives a comprehensive

survey

Let fL L g b e a realizer of an order P With every x P we asso ciate

k k i

the vector x x IR where x gives the p osition co ordinate of x in L

This mapping of the p oints of P to p oints of IR emb eds P into the comp onentwise

ordering of IR Ore dened dimP as the minimum k such that P emb eds into

IR in this way Since the pro jections of such an emb edding on each co ordinate

yield a realizer the two denitions are equivalent see Figure

CHAPTER INTRODUCTION

g t t

h t

f t

d e t t

t c

b t t

a t

a c f b d e g h

Figure A dimensional p oset P and an emb edding of P in the plane

Lemma A poset P X is two dimensional i there is a mapping

of the elements x X to intervals I on the real line such that x y i

I I

x y

Pro of We rst claim that there is no loss of generality if we only deal with fami

lies I of intervals containing a common p oint To see this note that for any

p ositive numb er M we may increase each interval in length by M symmetrically

ab out its center without aecting the containment relation

A geometric argument then proves the lemma Let r IR b e a p oint with r

I Take r as pivot and turn the left side of the axis up thus transforming

the intervals into ho oks The construction is illustrated in Figur It should

b e clear how to carry out the converse construction 2

Interval Orders

Let I b e a family of intervals on the real line This family may b e used to

dene a graph and two partial orderings on X

intersection graph Edges corresp ond to pairs of intersecting intervals That

is G X E is the intersection graph of the family if E f fx yg

I I g A graph obtained in this way is called an interval graph

x y

INTERVAL ORDERS

a e d b c a b c d e a b c d e

Figure A two dimensional order in two representations

containment order The ordering on X is given by prop er containment ie

x y i I I As we have seen in Lemma the class of p osets

x y

obtained in this way is exactly the class of dimensional orders

visibility order Here we put x y if when lo oking to the right every p oint

of I can see every p oint of I In other words x y i I is entirely to the

x y x

left of I A partial order obtained in this way is called an interval order

In each of the three cases the collection I is called an interval rep

resentation In many arguments we will have to refer to the endpoints of the

intervals corresp onding to the elements x X To simplify these references let us

adopt the convention that a b is the interval of x ie a is the left and b

x x x x

is the right endp oint

From the denitions we immediately obtain

Lemma G X E is an interval graph i it is the cocomparability

graph of an interval order That is there is an interval order I X

such that fx yg E i xjjy in P

With Predx fy X y xg we denote the set of all predecessors of x in

P X Dually Succ x fy X x yg denotes the set of all successors

of x

There are several imp ortant characterizations of interval orders we combine

some of them in the next theorem

CHAPTER INTRODUCTION

t t

j k

t t t

h i g

b d

A A

a e

A A

t t t A

f i k

d e f

A A

t t H A

b c

t T

Figure An example of an interval order with an interval representation In

most of our gures we indicate the intervals by rectangles

Theorem Interval Order Characterization

Let P X be a partial order The fol lowing statements are equivalent

P is an interval order

P does not contain a as induced suborder that is a subset

fu v x yg of X with u v x y and no more comparabilities see

Figure

For al l x y X the sets of predecessors are related by Pred x

Pred y or by Pred x Pred y ie the sets of predecessors are

linearly ordered with respect to inclusion

The sets of successors are linearly ordered with respect to inclusion

Pro of First let us insp ect that an interval order can not contain a Note

that in an interval representation of a chain x y we necessarily have b a

x y

Now let u v and x y b e two chains and consider the ordering of the

numb ers b a b and a From b a and b a we conclude that either

u v x y u v x y

b a or b a and hence u y or x v In any case the set fu v x yg

u y x v

can not induce a

To see that implies supp ose that do es not hold Then we nd v

and y such that there are elements u Predv and x Predy with x Pred v

and u Pred y The element set fu v x yg is thus recognized as a

Essentially the same argument can b e used to show that implies If

do es not hold we nd a with chains u v and x y The sets Succ u

INTERVAL ORDERS

and Succ x then are not related by inclusion The dual argument shows that

implies

We nally prove that implies Let P P P b e the chain of

sets of predecessors For x X let a f g b e such that Pred x P

x a

and let b b e the least index with x P if such an index do es not exist ie

if x MaxP we dene b Note that since x Pred x we always

have a b We claim that the op en intervals a b for x X represent

x x x x

P If x y then x Pred y hence b a and the interval of x precedes

x y

the interval of y On the other hand if xjjy then x Predy implies a b

y x

and y Pred x implies a b Therefore the intervals of x and y intersect

x y

This representation of an interval order by op en intervals with integer endp oints

is called the canonical representation 2

Of course if we would have applied the dual construction to prove that

implies the resulting interval representation would b e the same ie the

canonical representation As a consequence we obtain that the numb er of dierent

sets of predecessors agrees with the numb er of dierent sets of successors this

numb er denoted by is called the magnitude of the interval order

If we replace the op en intervals a b of the canonical representation by

x x

the p ossibly degenerate intervals a b we obtain a closed representation

x x

with endp oints in f g Therefore every nite interval order has op en and

closed representations In fact the magnitude of an interval order I is the least

p ositive integer n for which I has a closed representation with integer endp oints

in f ng

With an interval order I X we can asso ciate two sp ecial linear orderings

In the increasing Pred order we have x y i Predx Pred y or Pred x

Predy and Succ x Succ y elements with equal holdings ie elements with

identical sets of predecessors and of successors are ordered arbitrarily In the

decreasing Succ order we have x y if i Succ y Succ x or Succ y

Succ x and Pred y Pred x elements with equal holding are again ordered

arbitrarily Note that Pred order and Succ order are linear extensions In general

the Predorder of I and the Succ order of I need not b e the same

An interval order I X is called a semiorder i I has an interval

representation a b such that b a for all x X We close the

x x x x

intro duction with mentioning the characterization theorem for semiorders

CHAPTER INTRODUCTION

w u

v y v u u u

u x u u u u u

Figure The partial orders and

Theorem SemiOrder Characterization

Let I X be an interval order The fol lowing statements are equiva

lent

I is a semiorder

I does not contain a as induced suborder that is a subset

fu v w xg of X with u v w and no more comparabilities see

Figure

The Pred order and the Succ order of I are identical

References for Chapter

An Ian Anderson Combinatorics of Finite Sets Oxford University Press

Fi PC Fishburn Interval Orders and Interval Graphs Wiley NewYork

Mo RH Mohring Algorithmic asp ects of comparability graphs and inter

val graphs in Rival ed Graphs and Order Reidel Publishing Company

Mo RH Mohring Computationally tractable classes of ordered sets Al

gorithms and Order in Rival ed Reidel Publishing Company

Tr WT Trotter Combinatorics and Partially Ordered Sets Dimen

sion Theory John Hopkins Press

We DB West Parameters of partial orders and graphs packing covering

and representation in Rival ed Graphs and Order Reidel Publishing

Company

Chapter

The Jump Numb er of Interval

Orders

Intro duction and Overview

Let L x x x b e a linear extension of P Two consecutive elements

x x of L are separated by a jump i x is incomparable with x in P

i i i i

sometimes we call this the jump after x If x x the pair x x is called

i i i i i

a bump The total numb er of jumps of L is denoted by s L or sL The jump

number sP of P is the minimum numb er of jumps in any linear extension ie

sP minfs L L is a linear extension of P g

A linear extension L of P with s L sP will b e called optimal

The jump numb er has b een intro duced by Chein and Martin CM The

minimization problem determine sP and nd an optimal linear extension has

b een shown to b e NP hard even for bipartite orders Pulleyblank Pu Muller

Mu Nevertheless ecient metho ds for jumpminimization have b een found

for large classes of partially ordered sets such as Nfree orders Rival Ri cycle

free orders Duus Rival Winkler DRW or orders with b ounded decomp osition

width Steiner St In this chapter we consider the jump numb er problem on

interval orders

In the next section it is shown that the jump numb er problem can b e reduced

to a choice problem nd an appropriate rst element for an optimal lin

ear extension We then give some algorithms using dierent choice rules In

particular the greedy rule and some derivates of the greedy rule are investigated

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

The third section deals with lower b ounds for the jump numb er of interval

orders These b ounds are derived from a socalled auxiliary list which is a rep

resentation of interval orders which is close to a linear extension

The auxiliary list is used in section four for the analysis of two algorithms

First we study the Tgreedy algorithm Felsner Fe which has a p erformance

ratio of That is if P is an interval order and L is the linear extension of

s L Approximation P generated by the Tgreedy algorithm then sP

algorithms with the same p erformance b ound have recently b een prop osed by

Syslo Sy and Mitas Mi

We then give an algorithm based on bipartite matching which recognizes defect

optimal interval orders Mitas Mi obtained the rst p olynomial recognition of

defect optimal interval orders Earlier a sub class of defect optimal interval orders

has b een characterized by Faigle and Schrader FS

In general approximation algorithms are the b est we can hop e for This

results from the NP comp eteness of the jump numb er problem on interval orders

Mitas Mi In section ve we give a mo died pro of of her result which nicely

ts into the theory develop ed b efore

A dierent light is shed on the complexity of the jump numb er problem on

interval orders by a recent result of de la Higuera Hi He shows that the jump

numb er problem is p olynomial on the class of semiorders

Greedy Linear Extensions and Starting El

ements

In the context of the jump numb er problem it is useful to view linear extensions

as b eing constructed by the generic algorithm Algorithm on page Note

that the choice made for the i element determines whether the pair x x

i i

is a jump or a bump This suggests the idea of guiding the choices such that

elements pro ducing bumps are preferred This idea leads to the greedy algorithm

Algorithm

Linear extensions constructed by the greedy algorithm are called greedy linear

extensions Part of the interest in greedy linear extensions has its origin in the

following theorem

GREEDY LINEAR EXTENSIONS AND STARTING ELEMENTS

Algorithm

Greedy Algorithm

for i to n do

if MinP Succ x do

cho ose x MinP Succ x

i i

else

cho ose x MinP

L L x

P P n fx g

output L

Theorem Rival Zaguia

Every poset P has a greedy linear extension which is optimal

Pro of Call an element x nongreedy in L x x x if for Q P n

fx x x g we have MinQ Succx and x MinQ Succx

i i i i

Note that linear extensions without nongreedy elements are exactly the greedy

linear extensions

Now assume that P has no optimal greedy linear extensions For a linear

extension L of P let ngL b e the least index of a nongreedy element in L if L is

greedy let ngL n Our assumption implies

r maxfngL L optimal g n

Now let L y y y b e an optimal linear extension with ngL r Dene

L y y y as follows

y y for i r

Cho ose z Min P n fy y g Succ y and let y z

r r

If z y then y y for r i k

k i

y y for i k

The linear extensions L and L only dier in three consecutive pairs namely

y y y y and y y have b een replaced by y z z y and

r r r r

k k k k

y y At least two of the pairs in L are jumps the pair y y and

r r

k k

since y is minimal in P n fy y g also the pair y y On the other

k k k

hand at least the pair y z in L is a bump therefore sL sL and L

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

is optimal But ngL ngL r contradicting our assumption and hence

proving the theorem 2

Rival Ri proved that any greedy linear extension of P is optimal if P is an N

free order Since then a great amount of work has b een invested into investigations

of the interplay b etween greedy and optimal linear extensions Ghazal et al

GSZ for example characterized greedy interval orders ie interval orders with

the prop erty that any optimal linear extension is greedy

From the algorithms of this section and Theorem we conclude that the

optimality of a linear extension only dep ends on a go o d choice of x MinP

Faigle and Schrader FS dened the set Start P MinP of starting elements

of P as the set of go o d choices ie Start P fx P x is the rst element in

some optimal linear extension of P g

The following two lemmas are valid for the starting elements of arbitrary

partial orders If P X we will use the abbreviation P to denote the

induced order on the set X n fag

Lemma If a MinP and sP sP then a Start P

Pro of Take any optimal extension L of P Since a is a minimal element of P

the concatenation L a L is a linear extension of P Counting jumps proves

the optimality of L 2

Lemma If a Start P and sP sP then a has a successor which

is a starting element of P ie Succ a Start P

a a

Pro of Let L a b L b e any optimal linear extension of P starting with

a If b Succ a or equivalently if a b is a jump then sP s b L

s L sP This would contradict sP sP so b Succ a The

equalities sP sP sL sb L show the optimality of b L for P

a a

so b Start P 2

The next algorithm Algorithm is very similar to the greedy algorithm

The starty algorithm diers however from the former in that it never fails in

generating optimal linear extensions

Lemma If L is the linear extension of P which is generated by the

starty algorithm then s L sP ie L is optimal P

GREEDY LINEAR EXTENSIONS AND STARTING ELEMENTS

Algorithm

Starty Algorithm

for i to n do

if Start P Succ x do

cho ose x Start P Succ x

i i

else

cho ose x Start P

L L x

P P n fx g

output L

Pro of The pro of is by induction on jP j First note that if L a L is generated

by the starty algorithm with input P then with input P the starty algorithm

pro duces L wrt appropriate choices Comparing the jump numb er of L and L

we nd two p ossibilities

If sL sL then the trivial inequality sP sP and the optimality

of L for P imply the optimality of L

Otherwise ie if sL sL there is a jump after a hence Succ a

StartP and by Lemma sP sP Again the optimality of L is

a a

deduced from the optimality of L 2

Corollary The NP hardness of the jump numb er problem implies that in gen

eral the identication of starting elements is NP hard to o

We now turn to interval orders Here we have slightly more information on

starting elements than given in Lemmas and From the characterization

theorem for interval orders Theorem we know that in an interval order

I X the sets of successors of any two elements x y X are related by

either Succx Succ y or Succx Succ y In the sequel we will often

refer to the decreasing Succ order and esp ecially to the minimal elements in the

decreasing Succ order therefore we intro duce a new notation SMinI f x

X Succ x Succ y for all y Xg

Lemma Faigle Schrader Let I X be an interval order If

a MinI then a Start I i a SMinI or sI sI

Pro of In an interval order there can b e at most one starting element with

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

sI sI since only for the Succ maximal element which has to b e unique

in this case there may exist b Succ a MinI as required see Lemma

Any other a Start I thus fullls the equation sI sI

So it only remains to prove that every Succ maximal element a is a starting

element First note that if L x x x is optimal and a x then the

set fx x g is an antichain in I since Succ x Succ a for j i Let

i j

x b e the rst element of L with x x then k i and g L gL

k k

x x x x x is an op erator which do es not increase the numb er of jumps

k k

The linear extension g L is optimal and starts with a 2

The preceding lemma motivated Faigle and Schrader FS to oer the fol

lowing greedyheuristic for jump minimization in interval orders Construct a

greedy linear extension L extending the decreasing Succ order that omits jumps

whenever p ossible We again present an algorithmic version of this approach

Algorithm

Faigle Schrader Heuristic

for i to n do

if Succ x SMinI do

cho ose x Succ x SMinI

i i

else

cho ose x SMinI

L L x

I I n fx g

output L

Let L b e constructed by the previous algorithm Faigle and Schrader FS

asserted s L sI The factor of two however is not really correct Consider

this interval order

d i

b h

a c

The linear extension of I constructed by the previous algorithm has jumps

it is ajbjcjdjejfjgijh The optimal linear extension of I only has jumps it is

adjbejcfhjgi However with the metho ds intro duced in the next two sections

the real factor of their heuristic is easily seen to b e less than

BOUNDS FOR THE JUMP NUMBER OF INTERVAL ORDERS

Bounds for the Jump Numb er of Interval

Orders

The main to ol in this section will b e the auxiliary list LI of an interval order

I X This list is an almost representation of I ie the list enco des almost

all the information required to generate the canonical representation of I In

fact sometimes we can not recover the left endp oint of an interval For the jump

numb er problem however the list LI contains all the relevant information

That is if two interval orders I and J have isomorphic lists then their jump

numb er is the same We build up the list LI of I in two steps

Take the elements of I in decreasing Succ order see page We thus

obtain a linear extension x x x

We now include some additional information in this list First observe that

for every pair x x in the list one of the following three cases applies

i i

a Succ x Succ x

i i

Call this an jump

b Succ x Succ x and x Succ x

i i i i

This are the bumps of

c Succ x Succ x and x Succ x

i i i i

Call this a jump

At every jump x x of we now x a b ox containing the set

i i

N Succ x Minfx x g This gives the list LI

i i

The auxiliary list then lo oks like

LI L 2 L 2 L 2 L

N N N

Example

b h

a d

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

Given this interval order I we obtain LI a2 jb2 jc2 jdjehjfjg and I

d ef

ajbjcjdjehjfjg As indicated by the rules there are jumps of them are counted

Remarks

Note that if x x is a jump and we x x x x then the greedy al

i i i

gorithm would cho ose as the i element some memb er of N Succ x

Minfx x g

The List L is uniquely dened up to interchanges of elements having the

same sets of predecessors and successors

The linear extension could also b e generated by the FaigleSchrader

heuristic

The N are pairwise disjoint ie an element y may app ear at most once

in one of the sets Succ x Minfx x g

i i

If y N for some i then y is a minimal element of the remaining p oset

hence in there must b e a jump just b efore y

There is another characterization of the Iinvariant of jumps in

Namely n where denutes the magnitude of the interval order I

An equality involving and is n bumps

To investigate the p erformance of algorithms for our problem it turned out

to b e useful to investigate the eect of the choices made on and More

precisely if a minimal element x I has b een chosen then the and of the

lists LI and LI resp ectively are to b e compared In a rst application of this

technique we will derive lower b ounds for the jump numb er from the list version

of the starty algorithm Algorithm We will also refer to this algorithm as

list starty algorithm

Lemma The linear extension LI generated by the list

S S

starty algorithm is an optimal linear extension of I

Pro of With reference to Lemma it suces to show that there is a run of

the starty algorithm with input I that generates LI Let x x x

S S

BOUNDS FOR THE JUMP NUMBER OF INTERVAL ORDERS

Algorithm

Starty Algorithm List Version

L LI

while there is a rst 2 in L do

L L 2 L decomp ose L

J f x x L g

if N Start J do

cho ose n N Start J

L L n LJ

else

L L L

output

the element x is in SMinI so by Lemma we know x Start I and x is

a suitable rst choice for the starty algorithm

Supp ose the starty algorithm has chosen x x the rst i elements of

i S

Let J I n fx x g If x is in SMinJ then the starty algorithm may

i i

cho ose x This is due to Lemma Otherwise x will b e an element of

i i

N Succ x MinJ In this case our algorithm found the decomp osition

x x 2 LJ and selected x as a starting element Hence again x

i N i i

may b e chosen by the starty algorithm 2

As announced b efore we now lo ok for an expression of s in terms of

and We b egin with the intro duction of some new variables Let c count the

numb er of times the while lo op is rep eated in a run of the list starty algorithm

ie c is the numb er of b oxes the algorithm nds on its way through the list L

Let c count the numb er of times the condition N Start J app ears false

ie c is the numb er of b oxes kept empty

We now turn to the remaining c c b oxes ie to the b oxes lled with some

starting element n N In each of these cases the tail of L ie L LJ is

replaced by the new list LJ A close lo ok at the way auxiliary lists are built

enables us to characterize the transition LJ LJ Since n MinJ but

n SMinJ we know that in L the element n is preceded by an element x that

is incomparable with n If Succ x Succ n ie Succ x Succ n then in L

we nd a b ox b etween x and n and p ossible patterns for the transition are

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

x2 j n2 j y xy

N N

i i

x2 j n2 j y x2 j y

N N N N

i i i i

x2 j ny xy

x2 j n j y x2 j y

N N

Remark These transitions corresp ond to

Succ n Succ y y Succ n y Succ x

Succ n Succ y y Succ n

Succ n Succ y

Note that in the auxiliary list we gave preference to the element with maximal

set of predecessors in SMin In the last case ab ove we thus obtain Pred n

Pred y from the fact that n precedes y in L This excludes the transition

x2 j n j y xy

Now let b e the jump counters for L LJ and let b e those

for L n LJ Dep ending on the pattern of the transition LJ LJ

n n

we nd

in case

in cases and

in case

If Succ n Succx then the jump xjn b etween x and n is counted by

When n is pulled forward then with resp ect to the rest of the list x takes the

role of n Thus the new jump counters are

in case Succ n Succ x

Now partition the c c starting elements n which have b een pulled forward

by the algorithm according to the typ e of transition LJ LJ

Let c count the transitions of typ e

c count the transitions of typ e and

c count the remaining transitions

With these denitions the validity of the following equations is obvious

c c c c c

c c c

c c c c

At this p oint we can express sI s in terms of and the c

S i

AN APPROXIMATION ALGORITHM

Lemma The jump number of I is s c c c

Pro of In LI we distinguish b etween the jumps counted by and those counted

by If we transform LI step by step to ie for each of the c rep etitions

of the while lo op in the list starty algorithm we consider and observe which

jumps disapp ear We see that disapp earance only happ ens to jumps on the right

side of the current b ox Since numb er and typ e of disapp earing jumps dep end on

the transition LJ LJ only they are counted by the c Altogether we note

the disapp earance of c jumps counted by and c c jumps counted by

Hence s c c c insert the expression of given in

Formula to pro of the claim 2

We are ready for the b ounds now

o n

Theorem If I is an interval order then sI max

Pro of All the c are nonnegative therefore the rst b ound sI is an

obvious consequence of Lemma

Since c and c we may relax Equation to c c c c

With c we obtain c c c which is equivalent to c c c

2 This together with Lemma yields sI

An Approximation Algorithm

We start this section at the same p oint where the last one did end with some

calculations involving and Our rst aim will b e an algorithm with a

approximation factor Therefore it will b e useful to have a nicely expressible

lower b ound for sI

sI maxf g Lemma

Pro of In view of Theorem we only have to consider the second inequality

We distinguish two cases

If then

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

For reasons of an easy analysis we rst state the next algorithm in its list

version Algorithm

Algorithm

TGreedy Algorithm List Version

L LI

while there is a rst 2 in L do

L L 2 L decomp ose L

J f x x L g

cho ose n N

L L n LJ

output

When comparing this algorithm with the list starty algorithm we note the

following

Both algorithms start with LI and transform this list step by step into

their output

In each iteration of b oth algorithms some tail LJ of L is replaced by LJ

The p ossible transitions LJ LJ are the same in b oth algorithms ie

those of the previous section

Intro ducing the transition counters c we can thus express s in terms of

i TG

and some c

Lemma The jump number of is c c

Pro of The pro of is almost the same as for Lemma The only dierence is

that now each b ox found is also lled up hence c 2

sI Theorem s

Pro of In view of Lemma and the previous lemma we only have to show that

c c Since c c c see Equation and the c are

nonnegative this is obvious 2

It should b e evident that the Tgreedy algorithm can as well b e stated without

reference to the list LI For sake of completeness we also state this version

Algorithm

DEFECT OPTIMAL INTERVAL ORDERS

Algorithm

TGreedy Algorithm

for i to n do

if MinI Succ x do

if SMinI Succ x do

cho ose x SMinI Succ x

i i

else

cho ose x MinI Succ x

i i

else

cho ose x SMinI

L L x

I I n fx g

output L

Defect Optimal Interval Orders

The defect of a partially ordered set has b een intro duced in Giertz Poguntke

GP as def P jP j rankM where M is the incidence matrix of P ie

P P

M x y i x y otherwise M x y The main result of GP is

P P P

Theorem

sP def P

Remark Orders with equality ie with sP def P are called defect

optimal We now give a short and simple pro of for this b ound see Fe

Pro of Let L b e a jump optimal linear extension of P ie s P sP Order

the rows and columns of M according to the order of L Then M is an upp er

P P

diagonal matrix with a zero diagonal On the sup erdiagonal of M we nd a

for each bump and a for each jump x x of L see Figure Delete the

i i

rst column and the last row of M b oth have all entries as well as all the

rows and columns corresp onding to a on the sup erdiagonal The matrix M

thus obtained is an upp er diagonal All the entries on the diagonal of M are

hence rankM size M jP j sP With rank M rankM we

P P P

have nished the pro of 2

If I is an interval order then we may arrange the rows of M in Succ order

and the columns in Pred order This leads to a staircase shap e of M see Figure I

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

u e f a d c f b e

u u c

b d

A u u a c e

Figure An interval order I with its incidence matrix M Rows and columns

of M are arranged according to an optimal linear extension

In this representation M is easily seen to b e of rank where denotes

the magnitude of I Therefore the defect optimal interval orders are exactly the

interval orders with sI jIj the second equality can b e found in a

remark on page

a c d b f e

Figure The staircase shap ed M of the order given in Figure

From Lemma we know that for sI we need c c and

c in a run of the list starty algorithm This means that

All the b oxes found are lled

All the transitions LJ LJ are counted by c

Lemma A transition LJ LJ is counted by c i J J

n n

Pro of Let the elements x n y app ear consecutively in LJ With the transition

table it is easy to verify that a necessary and sucient condition for the transition

LJ LJ to b e counted by c is that either Succ n Succ x or Succ n

Succ y In b oth cases J J n

DEFECT OPTIMAL INTERVAL ORDERS

If the transition LJ LJ is not counted by c then Succ x Succ n

Succ y In LI the elements are in decreasing Succ order so the element n

contributes an indentation in the staircase shap e of the incidence matrix We

conclude that in this case J J 2

We now reformulate the ab ove conditions In a run of the list starty algorithm

we arrive at s exactly if the following holds

All the b oxes found are lled

For i let S b e the i largest set of successors of elements of I In

each set T f x Succ x S g there is at least one element that is not

i i

the n of any transition made by the algorithm

We now construct a bipartite graph G A B E as follows

As the vertices of A take the T for i

There are two kinds of vertices in B

Firstly there are jN j copies of each set N asso ciated to a b ox

j j

Secondly we take those elements of I which do not o ccur in any N

Let a b b e an edge of G

if a T and either b is a copy of some N with T N

i j i j

or b is a vertex of P and b T

This construction is illustrated in Figure

We now show how to reduce the question whether an interval order I is defect

optimal onto a matching problem in G

Theorem The jump number of I is i G has a matching of size

Pro of Supp ose sI then there is a run of the list starty algorithm satisfying

conditions and ab ove By condition we nd an x in each T which

i i

has not b een the n of any transition If x is a vertex of the second kind in

B then take the edge T x into the matching Otherwise x is an element of

i i i

some N use T b with a still unmatched copy b of N for the matching This

j i j

never causes trouble since exactly one n N gave rise to a transition for the

remaining elements of N the jN j copies of N suce Each of the many

j j j

T is nally matched i

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

a e

LI a2 jbdjc2 jehjfjg

cef

b d

a b c d e f g h

v v v v v

B B B

B B

v v v Bv Bv v B

a b d c e f c e f h

Figure An interval order and the asso ciated bipartite graph

Now assume that we have found a matching M E of size in G We use the

matching to construct a reduced list L First we lab el the edges of the matching

with elements of P The lab el a b of an edge a b is

i b if b is a vertex of P

iisome element of T N if a T and b is a copy of N

i j i j

Dene

N N n fa b a b is a matched edge and b is a copy of N g

j j j

Now let L b e obtained from L by exchanging each b ox 2 by 2 If we enter

R N

the list Tgreedy algorithm with L then condition is satised since each

N of L It remains set T contains a element x which do es not app ear in any

R i i

to show that we can ll all the b oxes to satisfy condition This can b e done

N j for all j 2 since j

The NP Completeness Pro of

In Theorem we have established two lower b ounds for sI In the last section

we saw that the question sI can b e decided in p olynomial time In

is NP complete contrast we will now prove that the decision sI

Again we start with extracting necessary and sucient conditions for equality

from the pro of of Theorem

THE NP COMPLETENESS PROOF

then c c and c Lemma If s

2 Pro of See the pro of of Theorem In addition we obtain c

The result will b e proved by a transformation from the NP complete problem

XC exact cover by sets For background information on the theory of NP

completeness and the XC Problem we refer to Garey and Johnson GJ The

transformation is due to Mitas Mi We start with a presentation of the XC

Problem

Exact Cover by sets

Instance A set Y of size q for some p ositive integer q together with a family F

of element subsets of Y

Question Do es F contain an exact cover of Y that is a subset F F of q pairwise

disjoint subsets of F

We now intro duce a construction which asso ciates an interval order I with

a given instance Y F of XC Let Y F consist of

Y fy y g n q

F fT T g T Y jT j

i i

The order I will consist of a basis representing the baseset Y together

with a Tsegment for each T F The interval representation of I is an op en

representation ie a representation by op en intervals

The basis B is a width interval order consisting of frame intervals a a

Y The intervals are and an interval y for each y

a a a

y k k

a n n a n n a n n

The Tsegment TS for T F consists of intervals The b o dy of the T

i i

i i i

segment consists of short intervals b b and c namely

i i

b n i j n i j and c n i n i

i i

The b o dy of the Tsegment is connected with the basis by three interval t t

and t representing the elements of T If T ft t t g and t y then

i i i i i ij

i i

the starting p oints of t and y coincide as well as the ending p oints of t and

j j

b that is j

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

t k n i j

Example For Y fy y y y y y g and F ffy y y g fy y y gg

the asso ciated interval order is shown in Figure

b b

a a y y b b c c b b

a a y y b b b b

a a y y

Figure The intervals of the rst T segment are dashed

As a useful consequence of this construction we can obtain the list LI

by concatenating the list LB with the lists LTS To give LB in a closed

form we need a further denition Let

g t y N fy g f t

ij k k

k j

We then have

LB a 2 ja 2 ja 2 jy 2 jy 2 j

N N N N N

y 2 jy 2 jy 2 jy a ja ja

n n fa g n fa g n

i i i i i i i i i

LTS 2 jb 2 jb jt 2 jb jt 2 jb jt c jb

i i i i

fb g fb g fb g fb g

We count B and B n Each Tsegment TS contributes TS

i i

and TS In the concatenated list LI we thus have m and

i YF

n m The lower b ound of Theorem gives sI m

m q With this we are able to state the main theorem of this section

Theorem The jump number of I is m q exactly if the XC

instance Y F has a solution

THE NP COMPLETENESS PROOF

Pro of Assume that the XC instance has a solution F Let ik and jk b e

t T F In a run of the Tgreedy algorithm cho ose such that y

jkik ik

N as the element to b e pulled into the b ox for all other b oxes cho ose t

the unique element contained in the b ox This leads to

j j j j j

j jy t jy t ja t ja t B a t

i i i i i

jy a jy a jy a y t

n n n n

i i i i i i i i i

c jt b jt b jt b jb TS b

TG i

i i i i i i

TS b jb b b jb c

TG i

The case applies if T is not used for the solution ie if T F otherwise the

i i

Tsegment of T is transformed to

The basis contributes n q jumps in B If T F then TS

TG i TG i

contains jumps In the q Tsegments corresp onding to T F only jumps are

in TS Altogether we have s q m q m q and

TG i TG

since achieves the b ound of Theorem it is optimal

For the converse direction we have to show how to derive a solution of Y F

from the fact that sI We may assume that an optimal linear

extension is pro duced by the list starty algorithm Here are prop erties of an

optimal run of this algorithm

While scanning the list the algorithm lls up each b ox it nds This b ecause

from Lemma we know that c

The algorithm may never cho ose y N to ll a b ox 2 The corre

k k N

sp onding transition would b e counted by c but c by Lemma

Therefore some t N is chosen

The claim that proves the theorem is the following In an optimal run of

the list starty algorithm from each set T F either al l three elements t

i i

t and t are pul led into their boxes in the base or al l three remain in the

corresponding Tsegment The elements chosen from the N therefore b elong

to only q dierent T which constitute an exact cover of Y

To prove the claim we rep eatedly apply the following argument

From Lemma we know that c Since each transition counted

by c removes an jump we may lo ok at jumps and decide which

of the two elements is pulled out

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

Rememb er that

i i i i i i i i i

c jb jt jb 2 jt jb 2 jt jb 2 jb LTS 2

i i i i

g fb g fb g fb g fb

Assume that t has b een pulled into the base When the algorithm nds 2

g fb

he has to use the element b to ll this b ox We then have the following situation

overbraced elements may have b een pulled out

z z

i i i i i i i i

c jb jt jb 2 jt b jb b

g fb

i i

Consider the jump b jt for this jump which is removed from the nal list

i i i

we can only remove jt has to b e pulled out From the jump b the element t

i i i i

t and this has to b e done Hence if t then also t and t have b een pulled into

the base

i i

Now assume that t did remain in the Tsegment but t has b een pulled out

After lling the rst two b oxes with the unique choice we remain with

i i i i i i i i i

c jb jt b jt jb b jb b

i i

Now the jump c jb cant b e removed contradicting the optimality condition

i i i

Therefore if t remains then t also remains Finally assume that t has b een

i i

pulled out Taking b forward into the b ox after t we would generate a transition

all counted by c contradicting the conditions of Lemma Hence with t

three t remain in the Tsegment 2

approximation This NP completeness result together with the existence of

algorithms motivates the following problem

does an approximation algorithm for Problem For which

the jump number problem on interval orders exist

References for Chapter

CM M Chein and P Martin Sur le nombre de saunts dune foret CR

Acad Sc Paris t serie A

DRW D Duffus I Rival and P Winkler Minimizing setups for cycle

free ordered sets Pro c Amer Math So c

REFERENCES FOR CHAPTER

FS U Faigle and R Schrader A setup heuristic for interval orders

Op er Res Letters

FS U Faigle and R Schrader Interval orders without o dd crowns are

defect optimal Computing

approximation algorithm for the jump numb er of Fe S Felsner A

interval orders Order

Fe S Felsner Bounds for the jump numb er of partially ordered sets

Preprint

GJ M Garey and D Johnson Computers and Intractability A Guide

to the Theory of NP Completeness Freeman San Francisco

GP G Giertz and W Poguntke Minimizing setups for ordered sets

a linear algebraic approach SIAM J on Alg and Discr Meth

GSZ T Ghazal A Sharary and N Zaguia On Minimizing the Jump

Numb er for Interval Orders Order

Hi C de la Higuera Computing the jump numb er on semiorders is

p olynomial Preprint

Mi J Mitas Algorithmen zur Berechnung der Semidimension sowie der

Jump Numb er von Intervallordnungen Diplomarb eit TH Darmstadt

Mi J Mitas Tackling the jump numb er of interval orders Order

Mu H Muller Alternating cycle free matchings Order

Pu W Pulleyblank On minimizing setups in precedence constrained

scheduling unpublished manuscript

Ri I Rival Optimal linear extensions by interchanging chains Pro c

Amer Math So c

St G Steiner On nding the jump numb er of a partial order by substi

tution decomp osition Order

Sy MM Syslo The jump numb er problem for interval orders a

approximation algorithm preprint

CHAPTER THE JUMP NUMBER OF INTERVAL ORDERS

Chapter

The Dimension of Interval

Orders

Intro duction and Overview

A set L L of linear extensions of P is a realizer of P if the intersection of

the L equals P This requires that all incomparable pairs xjjy are realized in

the following sense there are i j f rg with x y and y x The

L L

i j

dimension dimP of P is dened as the minimum size of a realizer for a survey

on dimension see KT By a theorem of Yannakakis Ya it is in general NP hard

to compute dimP For interval orders however the complexity of determining

the dimension is op en In this chapter we try to give a comprehensive survey of

the knowledge on dimension of interval orders

Although interval orders have a somewhat onedimensional nature their di

mension can b e arbitrarily high The rst pro of of this BRT was essentially

RamseyTheoretic Here we give the more recent argument of FHRT

For an integer n let I X denote the interval order whose elements

n n

are all the closed intervals with integer endp oints from n with i i j j

i i j We call the p osets in the family fI n INg canonical interval

orders

For integers n k with n k Erdos and Ha jnal EH dened the shiftgraph

and whose arc set consists Gn k as the directed graph whose vertex set is

of all pairs fx x x g fx x x g where fx x x g

k k k

In this context we always require the elements of a set fx x x g to b e

lab eled such that x x whenever i j In Section we will develop the chro

i j

CHAPTER THE DIMENSION OF INTERVAL ORDERS

matic numb er theory of shiftgraphs Here we need the following result proved

as Theorem b elow

Result The chromatic number of the double shiftgraph Gn satises

o log log log n Gn log log n

We are now able to give the lower b ound for the dimension of interval orders

Theorem Let n let I X be the canonical interval order

n n

and Gn be the double shift graph Then dimI Gn

Pro of Supp ose that dimI t and let L L b e a realizer of I Now

n n

dene a coloring t as follows For each fi i i g cho ose c t

such that i i i i in L We claim that is a prop er coloring of Gn

Supp ose on the contrary that fi i i g fi i i g c ie the edge

induced by the four element set fi i i i g is incident with vertices of the same

color Then in L we have i i i i i i Since i i i i in I

c n

this contradicts that L is a linear extension We conclude that is a prop er

coloring of Gn so dim I Gn 2

In the next section we will study upp er b ounds for the dimension of interval

orders First we intro duce the concept of marking intervals This is then used to

give common access to logarithmic b ounds in terms of the height Ra and the

width FM of interval orders

Then we investigate the step graph of an interval order and derive b ounds for

the dimension from arccolorings of the step graph

Finally we sketch the fascinating new construction of Furedi Ha jnal Rodl

and Trotter FHRT They prove that the dimension of a height n interval order

is b ounded by a function fn which is asymptotically equal to Gn

In Section we discuss chromatic and arcchromatic numb ers of directed

graphs and their line graphs The results are then applied to shiftgraphs giv

ing estimates of their chromatic numb ers A combinatorial interpretation of the

chromatic numb ers of shift graphs follows

Some Logarithmic Bounds

In this section we are going to develop upp er b ounds for the dimension of interval

orders These upp er b ounds are proved by giving a rule which generates a realizer

SOME LOGARITHMIC BOUNDS

L L of an interval order I such that the size r of the realizer is b ounded by

a function fpI where p is some parameter of interval orders The parameters

will b e the width the height and the staircase length see page of I The

main to ol in proving the rst two b ounds will b e an alternative denition of the

dimension of interval orders which relies on the following lemma

Lemma Let I X be an interval order with a closed representa

tion a b and L be a linear extension of I Then there is a function

x x

m X IR such that

mx a b

x x

if x y then mx my

A function m resp ecting is called a marking function A marking func

tion having prop erty with resp ect to L is called an Lmarking

Pro of For L x x an Lmarking is dened by

mx max a mx

i i

It is obvious that m resp ects prop erty By denition a mx To verify

prop erty it remains to show that mx b Note that for each i we nd

some l such that mx a Since L is a linear extension we get x x

i il i

hence a b 2

x x

il i

Conversely if m is a marking function of I then with x y i mx my

we obtain an extension Q X of I Moreover m is a Lmarking with

resp ect to every linear extension L of Q

Marking Functions and Bounds

A realizer L L of I X corresp onds to a family m m of marking

r r

functions such that for every incomparable pair xjjy there are dierent i j

f rg with m x m y and m x m y

i i j j

A sp ecial class of linear extensions is obtained if we concentrate on marking

functions with mx fa b g ie we cho ose one of the endp oints of the interval

x x

for the mark Such a marking function can b e transformed into a Bo olean vector

jXj

f f g by fx i mx a A family f f of Bo olean vectors

x r

gives rise to a realizer of I if for xjjy there are i j with f x f y and

i i

f x f y We now give two constructions for such a family of Bo olean

j j

vectors

CHAPTER THE DIMENSION OF INTERVAL ORDERS

The rst construction is essentially the construction of Rabinovich Ra Let

I b e an interval order of height h For k h dene H as the set of elements

of I with heightx k Note that each H is an antichain Moreover if k l

and x H y H then a a

x y

k l

We now lo ok for linear extensions L L of I with the following prop erties

Each L corresp onds to a Bo olean vector f in the sense given ab ove

i i

If x y H then f x f y for i s ie all the elements of

k i i

H are treated in the same manner

For an incomparable pair xjjy with x H y H and k l there

k l

is some f with f x and f y ie y precedes x in L

i i i i

A family L L of linear extensions with prop erty can b e extended to a

realizer with a single additional linear extension L An appropriate L is obtained

if the elements of H precede the elements of H in L for k l and the order

k l

of elements of the antichain H in L is exactly the inverse of the order of these

elements in L

By prop erty we only have to investigate families of vectors with f

f g Prop erty then is

k l

for k l there is some i with f and f

i i

Therefore if we arrange the vectors f as the rows of a matrix F then every pair

k l

f f of columns of F diers in some comp onent Since F is an s h matrix we

conclude that h

Now let f f b e dierent vectors of length s dlog h e We assume

that these vectors have b een arranged in reversed lexicographic order to form

the columns of a matrix F ie for k l h if i is the least comp onent

k l k l

with f so the columns of f then f and f This is prop erty

i i i i

F give a family of linear extensions with prop erties and we have proved

Rabinovichs theorem

Theorem The dimension of an interval order I of height h is bounded

by dlog h e

Supp ose Y and Z are any two disjoint subsets of X An injection of Y by

Z is a linear extension L such that z precedes y in L if y Y z Z and

yjjz Rabinovichs pro of of Theorem made essential use of the following

characterization of interval orders

SOME LOGARITHMIC BOUNDS

Theorem P X is an interval order i there is an injection of Y

by Z for every two disjoint subsets Y and Z of X

Pro of If P is not an interval order then there is a in P let fy z g and

fy z g b e the two chains The injection of fy y g by fz z g will not exist

For the converse consider a marking function m with my b and mz

a for all y Y and z Z 2

In the previous construction we partitioned the elements of I into the an

tichains H This time we will make use of a partition into chains If C and C

are disjoint chains in an interval order I then all the incomparabilities xjjy with

x C and y C can b e realized with two linear extensions L and L The

order of L corresp onds to the Bo olean function fx if x C and fy

if y C For L interchange the roles of C and C

Let C C b e a partition of I into disjoint chains By Dilworths theorem

we may assume that w width I If we decide to treat all the elements of C

the same a Bo olean vector f f g corresp onds to a linear extension of I A

family f f of Bo olean vectors f f g gives rise to a realizer of I if

for every two comp onents k and l there are i and j such that f

l k l

f and f f

j j i

Let f f b e a realizer of I in the ab ove sense and arrange the vectors f

k l

as the rows of a matrix F of size r w By every pair f f of columns of F is

an incomparable pair of elements in the Bo olean lattice B ie the columns are

w an antichain in B With Sp erners theorem we obtain

e d

Now let f f b e an antichain in the Bo olean lattice B and arrange the

f as the columns of a matrix F The rows f f of F then corresp ond to a

realizer of I This proves the theorem

Theorem The dimension of an interval order I is bounded by r if r

w is the least integer with

e d

Remark We will frequently need this function in the sequel therefore we in

tro duce a name for it Let

Nw minfr IN wg

d e

Combining the ideas of the previous constructions we can prove another the orem

CHAPTER THE DIMENSION OF INTERVAL ORDERS

Theorem The dimension of an interval order I which does not contain

a t is bounded by Nt

Pro of Let H b e the set of elements x of I with heightx k The forbidden

sub order guarantees that x y if x H and y H with l k t For

k l

k t dene H as the union of the levels H with We will

in the same manner All inequalities in I can treat all the elements of each H

b e realized if we take the linear extensions corresp onding to a set f f of

vectors in f g such that

l k l k

f and f f for k l there are i j with f

j j i i

together with an appropriate L which reverses the order in in each H relative

to L Since prop erties and are equivalent a copy of arguments yields the

t ie r Nt 2 condition we have to put on r namely

e d

Remark As a consequence of this theorem we obtain another result of Rabi

novich Ra The dimension of a semiorder is at most

The StepGraph and More Bounds

Since we will b e concerned with b ounds for the dimension of interval orders in

this section to o we may start sp ending a xed numb er of linear extensions and

then investigate how this set can b e augmented to give a realizer Our initial set

consists of two linear extensions Let a b b e the canonical representation of

x x

I X then

L takes x b efore y if b b or b b and a a

up x y x y x y

L takes x b efore y if a a or a a and b b

down x y x y x y

Let xjjy b e an incomparability which is not realized by L L if L takes x

up down down

b efore y then a a b b A pair of elements x y of I with this ordering

x y x y

of endp oints in the canonical representation will b e called a step in I

Figure An example for a step

if x and y have identical intervals then if x is b efore y in L i y is b efore x in L

up dow n

SOME LOGARITHMIC BOUNDS

A family L L of linear extensions of I extends L L to a realizer of I if

s up down

every step is reversed ie if there is an L taking y b efore x

The step graph SI X U of an interval order I X is the directed

graph with arcs y x if x y is a step of I A directed path in SI corresp onds

to a sequence of steps ie a staircase Let x x x b e a path in SI

then the left endp oints of the intervals of the x are ordered by a a a

i l

This proves the next lemma

Lemma The step graph SI of an interval order I is acyclic

Let F F b e a partition of the arcs of SI such that I F is acyclic for

every i We then can take linear extensions L of I F and obtain a realizer

i i

L L L L of I We now give a necessary and sucient condition on a

up down s

class F such that I F is acyclic A class F is called semitransitive if for every

path x x x in F the transitive arc x x is in SI

l l

Lemma I F is acyclic exactly if F is semitransitive

Pro of Let x x x b e a path in SI with x x Along the

l l

path we have decreasing endp oints a a a and b b

b Therefore it is imp ossible that one of the intervals a b and a b is

l l l

contained in the other The reason for x x then has to b e a comparability

From a a we conclude x x If F is a class containing the ab ove path then

l l

F I contains the cycle x x x x

Now let F b e a semitransitive class and assume that x x x is a

cycle in I F In this cycle let x x b e a sequence of Farcs together

i ij

with the closing comparability ie x x and x x By

ij ij i ij

the semitransitivity of F we have x x in SI hence a b The

x x

i ij

comparability gives b a hence a a Let y y y b e

x x x x

ij ij i ij

the subsequence of elements of x x x which app ear as the right hand side

of a comparability x x The starting p oints of the intervals in this sequence

j j

are strictly increasing ie a a a a This contradicts the

y y y y

assumption 2

Note that if F is semitransitive and x y is the transitive arc of some path

in F then every linear extension of I F will take x b efore y

CHAPTER THE DIMENSION OF INTERVAL ORDERS

Remark Coverings of the arcs of a digraph D X A by semitransitive classes

and by transitive classes ie partial orders are equivalent problems The mini

mal numb er of partial orders on X whose union covers all arcs of D gives a notion

of digraph dimension Unfortunately little seems to b e known ab out this concept

of dimension which gives a measure for the intransitivity of D

An arccoloring of a directed graph is an assignment of colors to arcs such

that consecutive arcs obtain dierent colors

Lemma An arccoloring of SI with k colors leads to a realizer of I

with k linear extensions

Pro of Let F F b e the color classes of a prop er arccoloring No F do es

k i

contain consecutive arcs hence F is semitransitive and by the previous lemma

I F is acyclic Let L for i k b e a linear extension of I F L L

up down

i i i

and the k linear extensions L together are a realizer of I 2

We now discuss two metho ds leading to arccolorings of SI They lead us to

two new b ounds for the dimension of interval orders

In Section Lemma we will prove that the arcchromatic numb er of

a digraph D is at most N D Therefore we can b ound the arcchromatic

numb er of SI by estimating the chromatic numb er

Let indx denote the indegree of the vertex x in SI and let indSI

max indx Using the well known list coloring algorithm with a list given by

a top ological ordering of the vertices of SI we obtain a coloring with at most

indSI colors With Lemma we then obtain

Theorem The dimension of an interval order I is bounded by the

expression NindSI

Here is a second strategy leading to an arccoloring of SI Lab el an element

x X with the length of the longest directed path in SI which ends in x

Let hx b e the lab el of x and h max hx Now assume that an arc

coloring of the transitive tournament T on h p oints is given The arcset of

T is fi j i j f hg i jg Assigning to x y SI the color of

hx hy T we will obtain a legal coloring of SI Therefore the arc

chromatic numb er of T is an upp er b ound for the arcchromatic numb er of SI h

SOME LOGARITHMIC BOUNDS

Arcchromatic numb ers will b e discussed later in this chapter Lemma there

we will prove that the arcchromatic numb er of T is dlog he

Since h is just the maximal staircase length of I we obtain

Theorem The dimension of an interval order I is bounded by the

expression

dlogmaximal staircase length of I e

Op en Problems

We have obtained b ounds for the dimension of an interval order I in terms of the

height of I the width of I the minimum t such that I do es not contain an t

and in terms of parameters of the step graph For each of this b ounds there are

interval orders for which the b ound dominates the others Unfortunately there

remain interval orders for which all these b ounds are p o orly bad We now present

such an example

Example Let In m b e the order dened by all the op en intervals of length m

with endp oints in n For this order only the b ound referring to m is go o d it

gives the true value dimIn m If we add all the op en length intervals

to obtain In m this b ound also go es up to m If C Y is a chain

P X is an arbitrary p oset and Q X Y induces P on X and C on

P Q

Y then the dimension of Q do es not exceed the dimension of P by more than

Therefore dimIn m In fact dim In m we leave this as an

exercise

This example shall serve as a motivation for the following op en problems see

FHRT

Problem Given an interval order I is it NP complete to determine

the dimension of I

Problem Is it true that for every n IN there exists some t IN so

that if I is an interval order with dimI t then I contains a subposet

Q which is isomorphic to the canonical interval order I n

CHAPTER THE DIMENSION OF INTERVAL ORDERS

The Doubly Logarithmic Bound

We are now ready for a sketch of the most ingenious construction in the eld It

is due to Furedi Ha jnal Rodl and Trotter see FHRT

Their work splits into two parts First they pro of an estimate for the di

mension of canonical interval orders Their b ound is asymptotically optimal ie

coincides with the lower b ound Theorem In the second part they pro of that

the dimension of a height n interval order is b ounded by the size of the realizer

constructed in the rst part for I

Although I did not contribute to the results of this section pro ofs are included

since they are a very surprising combination of ideas we have seen earlier in this

chapter

Canonical Interval Orders

Let I X b e a canonical interval order We again start with two sp ecial

n n

linear extensions L and L Note that this time we have to change the

up down

denitions slightly since we dened them for the case of op en intervals But I

consists of closed intervals We remain with the job of reverting the steps ie

the pairs x y with a a b b Now let t Nlog n ie if s

x y x y

e d

then n We will show that semitransitive classes F F F suce to

cover the arcs of SI This will prove the theorem

Theorem dimI Nlog n

m l

of B and let Let M M M b e the antichain of subsets on level

n s

f f f b e dierent column vectors in f g we assume that these functions

are in lexicographic order ie for k l if i is the least comp onent row with

k l k l

f f then f and f

i i i i

Lemma Lexicographic Prop erty If S n with jSj and i is the

least component in which the vectors ff k Sg are not identical then

l l

there is a k S such that for l S f for l k and f for l k

i i

Pro of This is an easy consequence of the lexicographic ordering 2

THE DOUBLY LOGARITHMIC BOUND

Let y x b e an arc in SI ie a a b b The distinguishing

n x y x y

a a b b

x y x y

row of y x is the least comp onent i in which the vectors f f f f

are not all identical we will denote this as dy x i It is an immediate

consequence of the lemma that we may classify an arc y x as balanced zero

dominant or onedominant by the following scheme

a b

y y

a b

x x

f f y x is balanced if i dy x and f f

i i i i

a b

y y

a b

x x

f f y x is zerodominant if i dy x and f f

i i i i

a b

y y

a b

x x

f f y x is onedominant if i dy x and f f

i i i i

If y x is zerodominant then let j b e the least comp onent in which the

a a b

x y x

vectors f f f are not all identical We call j the tiebreaking row and

write j tby x Of course dy x tby x

If y x is onedominant then let j b e the least comp onent in which the

a b b

y x y

vectors f f f are not all identical We call j the tiebreaking row and

write j tby x Again dy x tby x

We now dene a lab eling of arcs with y x f tg The classes

F F F are then dened via this lab eling y x F

If y x is balanced we let y x

If y x is zerodominant i dy x and j tby x we cho ose

some c M n M and let y x c

i j

If y x is onedominant i dy x and j tby x we cho ose

some c M n M and let y x c

j i

Lemma Each of the classes F c t is semitransitive

Pro of In fact we will pro of that F is transitive and no class F c t

do es contain consecutive arcs ie they are prop er arc color classes in SI

We rst deal with F Let z y and y x b oth b e balanced As arcs in

the step graph they are steps and we have a a b b and a a

x y x y y z

a a

y y

b b Note that since they are balanced we have f f and

y z

dyx dzy

b b

y y

f f Therefore dy x dy z let d dy x

dyx dzy

CHAPTER THE DIMENSION OF INTERVAL ORDERS

If a b we have an arc z x in SI We claim that z x is also

z x n

a b

x x

balanced Since f f we have dz x d If dz x d then z x

d d

is balanced

Otherwise assume d dz x d In the distinguishing row d of z x

b a b a

z z x x

This contradicts either dy x d or f or f f we have either f

0 0 0 0

d d d d

dz y d

The remaining case is b a ie x z Recall that d dz

x z

y dy x and note that d is the rst comp onent in which the vectors

a a b a b b

x y x z y z

f f f f f f are not all identical In the distinguishing row we then

b a

y y

a a b b

z x z x

f f which violates the f nd the pattern f f f

d d d d d d

lexicographic prop erty Therefore if x z is not a step we cannot have b oth of

z y and y x in F

Now let y x b e in F c t Our claim is that z y is not in F Let

c c

d dy x and d dz y

a b

y y

Assume that y x is zerodominant then f f therefore

d d

If z y is balanced then it is not in F

a b

y y

If z y is onedominant then f f and d d As x y is

0 0

d d

zerodominant c y x M but z y M since z y is

d d

onedominant Hence c z y

a b

y y

If z y is also zerodominant then f f and necessarily

0 0

d d

d d In the tiebreaking row t tbz y however the lexicographic

a b

y y

prop erty ensures f f therefore t d By denition c

t t

y x M and z y M n M Hence c z y

d d d

Now assume that y x is onedominant

If z y is balanced then it is not in F

If z y is zerodominant then t tby x tbz y is the rst

a b

y y

co ordinate with f f By denition c y x M n M

t t

while z y M n M Hence c z y

a b a b

y y y y

If z y is also one dominant then f f and f f

0 0

d d d d

hence d d The tiebreaking co ordinate t tby x is the rst

THE DOUBLY LOGARITHMIC BOUND

a b

y y

with f f therefore t d By denition c y x

t t

M n M while z y M M Hence c z y

t t

d d

General Interval Orders

For an interval order I with height I n we now dene a mapping I I

Let A A A b e an ordered sequence of maximal antichains covering all the

is the rst and A is the elements of I For x X let x if A

x x

last antichain in this sequence containing x Note that

implies x y

x y

x y implies

x y

Let F F F b e the family of semitransitive classes constructed in The

orem to cover the arcs of SI We have seen that dimI t

n n

fx y SI x y F g Now dene F

we have a family of semitransitive classes Lemma With F F F

of arcs of SI

Pro of If c the class F do es not contain consecutive arcs This prop erty is

transferred to F hence for c the class F is semitransitive

c c

For the case of c note that in F we never have an arc y x with

y x

This is true since f f prevents y x from b eing balanced

y x

With this additional information F is easily seen to b e semitransitive as well 2

It remains to take care for those incomparabilities which are realized by the

two linear extensions L and L of I as well as for the steps x y with

up down n

All this can b e done with the two linear extensions

y x

L takes x b efore y if or and

x y x y x y

L takes x b efore y if or and

x y x y x y

down

As a consequence we obtain

Theorem If I is an interval order with heightI n then dimI t

if t Nlogn

CHAPTER THE DIMENSION OF INTERVAL ORDERS

Colorings of Digraphs and ShiftGraphs

In the rst part of this section we develop a theory of colorings for directed graphs

In Lemma we exhibit relations b etween the chromatic numb er and the

arcchromatic numb er As an application we obtain estimates for the chromatic

numb ers of shiftgraphs

In the second part we prove a combinatorial interpretation of the chromatic

numb ers of shiftgraphs Theorem

I guess that all the material presented here b elongs to the folklore of the eld

parts of it can b e found in HE and FHRT However when I rst heard of

the Erdos and Ha jnal result on the chromatic numb ers of shiftgraphs Theorem

it to ok me some time to repro duce their estimates The techniques I used

are given in the rst part

Colorings of Directed Graphs

An arccoloring of a directed graph is an assignment of colors to arcs such

that consecutive arcs obtain dierent colors The arcchromatic number AD

of a digraph D is the minimal numb er of colors in an arccoloring of D The

chromatic number D of a digraph D is dened as the chromatic numb er of

the underlying undirected graph G ie D G With the next lemmas

D D

we exhibit some connections b etween AD and D

Lemma For every digraph D chromatic and arcchromatic numbers

are related by AD dlog D e

Pro of Given an arccoloring of D with l colors we will construct a vertexcoloring

with colors As colors for the vertices we use b o olean vectors of length l

i th i

Let c denote the i comp onent of the color of x and dene c i there is

x x

an arc y x of color i

Assume that two vertices u v of the same color are connected by an arc

v u Then the color of the arc v u has to app ear as the color of some arc

w v conicting with the prop er arccoloring 2

With the next lemmas we investigate converses of Lemma We b egin with

a sp ecial case which turns out to b e paradigmatic

COLORINGS OF DIGRAPHS AND SHIFTGRAPHS

Lemma Let T be a transitive orientation of the complete graph K

n n

ie a transitive tournament then AT dlog n e

Pro of Since T K n we obtain AT dlog n e from the previous

n n n

lemma

For the converse let n and let the arcs of T b e given by the pairs

i j with i j and i j f g Assign color m to all the arcs i j

m m

with i and j The uncolored arcs then induce two indep endent

subtournaments on p oints each The rst consist of all arcs i j with i j

m m m

f g the second of all arcs i j with i j f g

By induction we can arccolor each of them with colors from f m g We

using m colors 2 obtain a prop er arccoloring of T

Lemma Every graph G V E admits an acyclic orientation D so

that AD dlog G e

Pro of Let G k and let c V k b e an optimal coloring of G In D

an edge fu vg E is oriented as u v i cu cv We use an arccoloring

of the transitive tournament T with dlog k e colors Lemma to color u v

with the color of cu cv in T Therefore AD d log G e

k G

From Lemma we obtain the converse inequality 2

Lemma For every digraph D chromatic and arcchromatic numbers

are related by AD N D

Pro of Let G k and let an optimal coloring c V k b e given If

k then let M M M b e an antichain of subsets t Nk ie

e d

l m

on level of B To an arc u v assign as color some element from the set

M n M Consecutive arcs u v and v w obtain dierent colors since the

cv cu

color of u v is element of M while the color of v w is not in M 2

cv cv

The line graph LD of a digraph D V A is the directed graph with

vertex set A and arcs a b a b A if heada tailb ie a u v and

b v w for vertices u v w V From this denition we immediately obtain

Lemma If D V A is a digraph and LD is the line graph of D

then LD AD

CHAPTER THE DIMENSION OF INTERVAL ORDERS

From the denition of shiftgraphs page the next lemma is immediate

Lemma The shiftgraph Gn is the transitive tournament T

Higher shiftgraphs can be obtained as line graphs ie Gn k

LGn k

We conclude this part with our main theorem on the chromatic numb er of

shiftgraphs

Theorem log n Gn k N log n

Pro of This is an easy combination of the preceding lemmas 2

Asymptotics

Here we give the results of some computations which are required to obtain the

Erdos Ha jnal theorem as a consequence of Theorem

Theorem ErdosHa jnal

a The chromatic number of the double shiftgraph Gn satises

o log log log n Gn log log n

b The chromatic numbers of higher shiftgraphs satisfy

Gn k o log n

Pro of From Theorem and Remark b elow we obtain that Gn

is the least integer t with n that is t Nlog n

Note that by Sterlings formula

This and the fact that with n Nm we certainly have log n o log m gives

o log log m Nm log m

COLORINGS OF DIGRAPHS AND SHIFTGRAPHS

That is the inequality of a

Relax the previous formula to obtain

Nm o log m

and note that this implies

N m o log m

ie the conclusion of b 2

A Connection with Lattices of Antichains

A down set in an order P X is a set M X such that x M and y x

implies y M There is a onetoone mapping b etween down sets and antichains

given by M MaxM and A A fx P x y for some y Ag Due

to a well known theorem of Dilworth the down sets antichains of an order P

form a lattice denoted by AP The order relation of this lattice is given by set

inclusion ie if M N AP then M N i M N Note that we can view

A P AP as an op erator which maps p osets to p osets Let A P denote

the p oset resulting from k applications of A

The next theorem gives a surprising connection b etween the chromatic num

b ers of shiftgraphs and the size of certain down set lattices The case k

of the theorem can b e found in FHRT Let t denote the t element antichain

ie the p oset consisting of c pairwise incomparable elements

Theorem For al l integers n and k the shiftgraph Gn k is colorable

with t colors if there are at least n elements in the poset A t

Remark Note that in the case of k this theorem will give an alternative

pro of for Lemma

Pro of Supp ose that a prop er coloring of Gn k is given by c t Let

C x x x cfx x x g

k k

C x x fC x x x for some x x g

k k

and C b e the down set generated by C in At For j k we iteratively

dene

C x x fC x x x for some x x g

j j k j j j k j j

CHAPTER THE DIMENSION OF INTERVAL ORDERS

where C is the down set generated by C in A t

j j

d d

Next we show that C x C y for all k x y n Therefore all

k k

these sets are distinct Supp ose on the contrary that k x x n and

k k

d d

C x C x From this we conclude the existence of some x

k k k k k

d d

x with C x x C x x and iterating this argument we

k k k k k k k

c c

nd x x x such that C x x C x x ie

k k k

C x x C x x We now let d C x x x from

k k k

d C x x C x x we obtain the existence of some x x

k k

such that C x x x d This contradicts the assumption that c is a

prop er coloring

We close this direction of the pro of with the observation that at the b ottom

of A t there is a chain of k empty elements which can not o ccur as

C x for x k These elements are fg ffgg ff fgg g the last

element in this list is an enclosed by k pairs of braces

For the converse direction assume that A t contains s n elements and

let C C Cs b e a linear extension of A t ie if x y we never

have Cx Cy Therefore we can cho ose a down set Cx y in A t

which is an element of Cy n Cx We claim that if w x then there is an

element Cw x y Cx y n Cw x otherwise the down set Cx y would b e

contained in the down set Cw x but Cw x Cx and Cx y Cx

Rep eating this we can asso ciate a set of colors Cx x x with each

k subset of n such that Cx x x Cx x n Cx x

k k k

Finally we nd a color Cx x x Cx x n Cx x This

k k k

coloring of the k element subsets of n is a prop er coloring of Gn k 2

Remark The problem of counting the antichains in At B is a classical

one The estimates assert that the numb er of antichains in B is approximately

the numb er of subsets of the largest antichain ie

References for Chapter

BRT KB Bogart I Rabinovich and WT Trotter A b ound on

the dimension of interval orders Journal of Comb Theory A

REFERENCES FOR CHAPTER

EH P Erdos and A Hajnal On the chromatic numb er of innite

graphs in Erdos and Katona Eds Theory of Graphs Academic Press

FM S Felsner and M Morvan More b ounds for the dimension of

interval orders Preprint TU Berlin

Fi PC Fishburn Interval Orders and Interval Graphs Wiley NewYork

FHRT Z Furedi V Rodel P Hajnal and WT Trotter Interval

orders and shift graphs Preprint

HE CC Harner and RC Entriger Arc colorings of digraphs Journal

of Comb Theory B

KT D Kelly and WT Trotter Dimension theory for ordered sets in

I Rival ed Ordered Sets D Reidel Publishing Company

Ra I Rabinovich The dimension of semiorders Journal of Comb Theory

Ra I Rabinovich An upp er b ound on the dimension of interval orders

Journal of Comb Theory A

Ya M Yannakakis The comp exity of the partial order dimension prob

lem SIAM J Alg Disc Meth

CHAPTER THE DIMENSION OF INTERVAL ORDERS

Chapter

Coloring Interval Order

Diagrams

Intro duction and Overview

For a nonnegative integer k let I b e the interval order dened by the op en

k k

intervals with endp oints in f g It has height and is isomorphic to

the canonical interval order I see Chapter for canonical interval orders

Two vertices v and w in I are a cover denoted by v w exactly if the right

endp oint of the interval of v equals the left endp oint of the interval of w The

diagram D of I is thus recognized as the shift graph G see Chapter

I k

for shift graphs In general we denote by D the diagram of an interval order I

and we denote the chromatic numb er of the diagram by D

We again include a pro of for the next lemma since we will need similar metho ds

in later arguments for an alternative formulation and pro of of this lemma see

Lemma

D d log heightI e k

I k

Pro of Supp ose we have a prop er coloring of D with colors f cg With

each p oint i asso ciate the set C of colors used for the intervals having their right

endp oint at i Note that C For i j we have C C otherwise

j i

the interval i j would have the same color as some interval l i This proves

k c k

that all of the subsets C of f cg are distinct therefore and

c k

CHAPTER COLORING INTERVAL ORDER DIAGRAMS

A coloring of D using k colors can b e obtained by the following construction

Take a linear extension of the Bo olean lattice B and let C b e the i set in this

k i

list Assign to the interval i j any color from C n C A coloring obtained in

j i

this way is easily seen to b e prop er 2

We derive a lemma for later use and a theorem from this construction

Lemma In a coloring of D which uses exactly k colors every point

i f g is incident with an interval of each color

Pro of The crucial fact here is that every subset of f kg is the C for some

i Now cho ose any i f g and a color c f kg we have to show

that an interval of color c is incident with i

If c C then this is immediate from the denition of C Otherwise ie if

i i

c C then there is a j i such that C C fcg and the interval i j is

c c

i j i

colored c 2

With the next lemma we improve the lower b ound There are interval orders

I with D log heightI Compared with Lemma this is a minor

improvement but we feel it worth stating since later we will prove an upp er

b ound of log height I on the chromatic numb er of the diagram of I

such that Lemma For each k there is an interval order I

D dlog height I e k

as the order obtained from I see Lemma by removing the Pro of Take I

intervals of o dd length ie the interval order dened by the op en intervals i j

o n

k k

and j i mo d The height of I is which with i j

is the height of I however as we are now going to prove a prop er coloring

of I requires at least k colors Note that two intervals i j and i j with

j i induce an edge in the diagram of I if either j i or j i

In I we nd an isomorphic copy of I consisting of the intervals i j with

b oth i and j o dd Call this the o dd I The even I is dened by the interval

k k

i j with i and j even Let C b e the set of colors used for intervals with right

endp oint i and let D b e the set of colors used for intervals with right

endp oint i From Lemma we know that if b oth the o dd and the even copy

only need k colors then the C and the D have to form linear extensions of

i i

INTRODUCTION AND OVERVIEW

the Bo olean lattice B Now dene C as the set of colors used for intervals

k i

C is exactly the with leftendp oint i From Lemma we know that

complement of C With the corresp onding denition D and D are seen to b e

i i i

D We complementary sets as well Note that a prop er coloring requires C

i i

therefore have C D A similar argument gives D C Altogether we nd

i i i i

that the C have to b e a linear extension of B with C C for all i This

i k i i

is imp ossible The contradiction shows that at least k colors are required 2

Now we turn to the upp er b ound which we view as the more interesting asp ect

of the problem

Theorem If I is an interval order then

D log heightI

Pro of In this rst part of the pro of we convert the problem into a purely

combinatorial one The next section will then deal with the derived problem

Let I V b e an interval order of height h given together with an interval

representation For v V let l r left op en right closed b e the corresp ond

v v

ing interval With resp ect to this representation we distinguish the leftmost

hchain in I This chain consists of the elements x x where x has the

h i

leftmost rightendp oint r among all elements of height i It is easily checked

that x x is indeed a chain Now let r r b e the right endp oint of x s

h i i

interval and dene a partition of the real axis into blo cks The i blo ck is

Bi r r

i i

This denition is made for i h with the convention that B extends to

minus innity and Bh to plus innity

In some sense these blo cks capture a relevant part of the structure of I This

is exemplied by two prop erties

The elements v with r Bi are an antichain for each i This gives a

minimal antichain partition of I

If r Bj then l Bi for some i less than j

v v

Supp ose we are given a sequence C C of sets of colors with the following

prop erty

CHAPTER COLORING INTERVAL ORDER DIAGRAMS

C C C for all i j h

j i i

A sequence with this prop erty will henceforth b e called an sequence The

sequence C C may b e used to color the diagram D with the colors

h I

o ccurring in the C The rule is to an element v V with l Bi and

r Bj assign any color from C n C C This set of colors is nonempty

j i i

by the prop erty of the sequence C since i j We claim that a coloring

obtained this way is prop er Assume to the contrary that there is a covering

pair w v such that w and v obtain the same color Let r Bk and l Bi

w v

Since w v we know that k i Due to our coloring rule we know that the

color of w is an element of C and the color of v is not contained in C C

k i i

hence k i This however contradicts our assumption that w v since

l Bi and l r r gives w x v

x v x

i i

We have thus reduced the original problem to the determination of the min

imal numb er of colors which admits a sequence of length h We will demon

strate in the next section Lemma and Lemma how to construct a

k j

using n colors This will complete the pro of of sequence of length

the theorem 2

k j

for the maximal In the third section we give an upp er b ound of

length of a sequence From the pro of we derive some further prop erties

sequences of this length necessarily satisfy Finally we apply the construction of

long sequences to the problem of nding long cycles b etween two consecutive

levels of the Bo olean lattice A famous instance of this problem is the question

whether there is a Hamiltonian cycle b etween the middle two levels of the Bo olean

lattice see eg KT Sa The b est constructions known until now could guar

antee cycles of length N where N is the numb er of vertices and c

We exhibit cycles of length N

A Construction of Long Sequences

Let tn k denote the maximal length of a sequence C of sets satisfying

C f ng

jC j k and

if i j then C C C

j i i

A CONSTRUCTION OF LONG SEQUENCES

Lemma

tn k

Pro of The sequences actually constructed will have the additional prop erty

jC C j k for all i

i i

The pro of is by induction For all n and k or k n the claim is obviously

true

Now supp ose that two sequences as sp ecied have b een constructed on

f n g First a sequence of ksets A A A of length s

and second a sequence of k sets B B B of length t

Prop erty guarantees that there is a p ermutation of the colors such that

A B B Now let

A for i s

fng for s i s t B

The length of the new sequence is s t Prop erties and

are obviously true for the sequence C and prop erty is true for b oth the A

and the B sequence These observations and the choice of give prop erty

for the C sequence It remains to verify prop erty If i j s this

prop erty is inherited from the A sequence If s i j it is inherited from

the B sequence In case i s j we have n C and n C C The

j i i

remaining case is s i j Here the choice of and the sacrice of B show

that C C A B fng B B fng Again prop erty can b e

s s s

concluded from this prop erty for the B sequence 2

For k and k n we can prove that the inequality of Lemma is

tight but in general the value of tn k is op en

Problem Determine the true value of tn k

Let T n denote the maximal length of a sequence C of sets satisfying

C f ng and

if i j then C C C

j i i

Lemma

n n

T n

k o dd

CHAPTER COLORING INTERVAL ORDER DIAGRAMS

Pro of Let Ln k b e the n ksequence constructed in the preceeding lemma

n with appropriate p er n L n L We claim that L L

mutations is a sequence of subsets of f ng The s can b e found

j k

recursively id and if has b een determined then is chosen as a

k k

p ermutation such that the last set of the sequence L n k is a subset

of the rst set of L n k Let C b e the i set in the sequence L We now

check prop erty If the three sets C C and C are in the same subsequence

i i j

k k

L n k then the prop erty is inherited from this subsequence If C L n k

C and C L n k with k k then jC C j is a consequence

j j i i

of prop erty for the subsequence L n k and gives the claim in this case

There remains the situation where C is the last set of its subsequence The

choice of the gives C C and the prop erty reduces to C C which is

k i i j i

obvious

The length of L is the sum over the length of the L n k used in L This is

k j

2 the sum over with k o dd which is

The Structure of Very Long Sequences

Theorem Let C C C be a sequence of subsets of f ng

k j

Then t

Pro of We start with some denitions For i t let

S f S C S C C g

i i i i

and s jS j Observe that with r jC n C j we have the equation

i i i i i

We now prove two imp ortant prop erties of the sets S

S S if i j

i j

Assume to the contrary that S S S and let i j From the denition

i j

of the S we obtain C S C C This contradicts with prop erty

i j i i

for the sequence C

THE STRUCTURE OF VERY LONG SEQUENCES

C S for all i

Assume C S If j i then C C gives a contradiction If j i

j i i j

note that C S from the denition If j i the contradiction comes

i i

from C C C

j i i

Therefore C and the S are pairwise disjoint subsets of B this gives the inequality

s t

We now partition the indices f t g into three classes

I fi jC j jC jg note that i I implies s

i i i

I fi jC j jC jg trivially s for i I

i i i

I fi jC j jC jg note that if i I then the corresp onding s is

i i i

jC jjC j

i i

relatively large ie s This estimate is a consequence of

Equation and the fact that C has to contain an element not contained

in C

We rst investigate the case I This condition guarantees that the sizes

of the sets in C is a nondecreasing sequence Since B has n levels the size

of the sets in C can increase at most n times ie jI j n and jI j t n

It follows that

X X

t s s

i i

iI iI

t jI j

t t n

k j

n n

in this case This gives t n Hence t

The case I is somewhat more complicated Let the numb er of descending

steps b e d and I fi i g Let m denote the numb er of levels the sequence

d j

is decreasing when going from C to C ie m jC j jC j and s

i i j i i i

j j j j j

Again we can estimate the size of I namely jI j n m

It follows that

X X X

t s s s

i i i

iI iI iI

CHAPTER COLORING INTERVAL ORDER DIAGRAMS

t jI j

d t t jI j jI j

d d

X X

d m d t t n

j j

Comparing this with the calculations made for the case I we nd that

k j

P P

d d

would require m d For each j t

j j

we have m Hence the ab ove inequality can never hold 2

j k

Remark Let T n b e the upp er b ound from the theorem We

have seen that a sequence C of length T n can only exist if I Moreover

the following conditions follow from the argument given for Theorem

There are exactly n increasing steps ie jI j n

If i I then s ie two consecutive sets of equal size have to b e a

shift C C n fxg fyg with x C and y C

i i i i

If i I then s ie if jC j jC j then there is an element x C

i i i i

such that C C fxg

i i

Every element of B is either an element of C or app ears as the unique

element of some S ie as C C

i i i

From this observations we obtain an alternate interpretation for a sequence

C of length T n in B In the diagram of B ie the nhyp ercub e consider

n n

the edges C C for i I and for i I the edges C T and T C

i i i i i i

where T is the unique memb er of S ie T C C This set of edges is a

i i i i i

Hamiltonian path in the hyp ercub e and resp ects a strong condition of b eing level

accurate After having reached the k level for the rst time the path wil l

never come back to level k see Figure for an example the bullets are

the elements of a very long sequence

Problem Do sequences of length T n exist for al l n

THE STRUCTURE OF VERY LONG SEQUENCES

f v v f

v f f v f v

v v v v

Figure A level accurate path in B

We are hop eful that such sequences exist Our optimism stems in part from

computational results The numb er of sequences starting with fg fg fng

is for n for n for n and there are thousands of solutions for

n The next case n could not b e handled by our program but Markus

Fulmek wrote a program which also resolved this case armatively

Long Cycles b etween Consecutive Levels in B

Let Bn k denote the bipartite graph consisting of all elements from levels k and

k of the Bo olean lattice B A well known problem on this class of graphs

is the following Is Bk k Hamiltonian for al l k Until now it was

known that this is the case for k Since the problem seems to b e very hard

some authors have attempted to construct long cycles The b est results see Sa

lead to cycles of length N where N is the numb er of vertices of

Bk k and c

Theorem In Bn k there is a cycle of length

n n

max

n k k

Pro of Note that the graphs Bn k and Bn n k are isomorphic it thus

suces to exhibit a cycle of length in Bn k To this end take a

CHAPTER COLORING INTERVAL ORDER DIAGRAMS

sequence C C of k sets on f n g From Lemma we

know that t can b e achieved Now consider the following set of edges

in Bn k

C fng C C fng for i t

i i i

C C fng C fng for i t

i i i

C fng C fn ng and C fn ng C fn g

t t t t

C fn g C C fn g for i t

i i i

C C fn g C fn g for i t

i i i

C fn g C fn ng and C fn ng C fng

The pro of that this set of edges in fact determines a cycle of length t in Bn k

is straightforward 2

With a simple calculation on binomial co ecients we obtain a nal theorem

N Theorem There are cycles in Bk k of length at least

References for Chapter

FHRT Z Furedi P Hajnal V Rodl and WT Trotter Interval

orders and shift graphs Preprint

KT HA Kierstead and WT Trotter Explicit matchings in the

middle two levels of the b o olean lattice Order

Sa C Savage Long cycles in the middle two levels of the b o olean lattice

SIAM Conference on Discrete Mathematics

Chapter

Interval Dimension and

Dimension

Intro duction and Overview

A family fQ Q g of extensions of a partial order P is said to realize P or

to b e a realizer of P i P Q Q ie x y in P i x y in Q for

k i

each i i k If we restrict the Q to b elong to a sp ecial class of orders and

seek for a minimum size realizer we come up with a concept of dimension with

resp ect to the sp ecial class

Interval dimension denoted idimP is dened by using interval extensions of

P Since linear orders are interval orders we obtain the trivial inequality

idimP dimP

It is well known that interval orders of large dimension exist see Chapter

Hence the gap b etween idimP and dimP may b e arbitrarily large

j j

Let I fI I g b e an op en interval realizer of P and let a b b e the

x x

interval corresp onding to x P in a xed representation of I We now nd a

j j

box embedding of P to IR With x P we asso ciate the b ox a b

x x

IR Each of these b oxes is uniquely determined by its upper extreme corner

k k

u b b and its lower extreme corner l a a Obviously

x x

x x x x

x y in P i u l comp onentwise The pro jections of a b ox emb edding onto

x y

each co ordinate yield an interval realizer so the concepts of b ox emb eddings

and interval realizers are equivalent For interval dimension the b ox emb eddings

thus play the role of the p oint emb eddings into IR intro duced by Ore for order

dimension

CHAPTER INTERVAL DIMENSION AND DIMENSION

A b ox emb edding do es not only dep end on the realizer I of P but also on

the representations of the I Now we dene the poset BI of extreme corners

asso ciated with a b ox emb edding or equivalently with an interval realizer I of

P The vertices of BI are l u for x P The order relation of BI is given

x x

by the comp onentwise order in IR

By denition we have an emb edding of BI in IR so dim BI k idimP

The starting p oint of our investigations was the following question concerning the

interplay b etween dimension and interval dimension

Is dim BI idimP

In the next section we dene a transformation P BP such that idimP

dim BP We provide two interpretations of this transformation a combinatorial

one and a geometrical one In the combinatorial interpretation the elements of

BP are subsets of P For the geometrical one we present a normalizing pro cedure

I I for b ox emb eddings and nd BP BI From the pro ofs we obtain

an armative answer to the question ab ove

In the third section we then investigate several consequences of the main

result First we study the transformation P BP on sp ecial partial orders of

height In particular we show that the standard example S of an ndimensional

order is an almost xed p oint of the transformation P BP in particular

dimS idimS

n n

Second we investigate the relationship with the split op eration Tr This has

a surprising consequence for the iterated transformation P BP B P

B P For every n there are partial orders P such that dimP

k n

dim B P for all k n but dimP dim B P m where m is arbitrary

Third we relate the interval dimension of sub divisions of P to the dimension

of P thus providing a theoretical framework for the examples of Spinrad Sp

Finally we show the comparability invariance of the transformation P BP

which as a consequence gives another pro of that the interval dimension is a

comparability invariant

Reducing Interval Dimension to Dimension

In the last section we dened the p oset BI of extreme corners asso ciated with

a b ox emb edding of P in IR We now show that BI inherits some structure

which is indep endent of the realizer I leading to the b ox emb edding

REDUCING INTERVAL DIMENSION TO DIMENSION

Lemma Let BI be the poset of extreme corners of a box representa

tion of a partial order P

a If the lower extreme corners of x and y are comparable in BI eg

l l then the predecessor sets of x and y in P are ordered by inclusion

x y

ie Pred x Pred y

P P

b If the upper extreme corners of x and y are comparable in BI eg

u u then the successor sets of x and y in P are ordered by reversed

x y

inclusion ie Succ x Succ y

P P

c If the lower extreme corner of x and the upper extreme corner of y

are related by l u then Pred z Pred x for al l z Succ y or

x y

P P P

Pred z equivalently Pred x

P P

zSucc y

Pred z Pred y d If u l then

x y

P P

zSucc x

j j

Pro of a From l l we obtain a a for all j Therefore in each I x has

x y

less predecessors than y ie Pred x Pred y The claim now follows from

j j

Pred x Pred x since the I realize P

P j j

b The pro of of this part is symmetric to part a

j j j j j

c From l u we have a b If z Succ y then necessarily a b a

x y

x y z y x

Hence Pred z Pred x for all j The claim follows

j j

d From u l we immediately obtain x y ie y Succ x therefore

x y

Pred z 2 Predy

zSucc x

All statments except the conclusion part of b use only the sets Pred x and

Pred z This irregularity is resolved with the next lemma

zSucc x

Predz if and only if Succ x Succ y Pred z Lemma

zSuccy zSuccx

Pro of The if direction is trivial We now prove the only if direction Let

z Succ x and note that y Pred z From the assumed inclusion

zSuccy

we obtain y Predz hence z Succ y 2

Denition With each vertex x of a partial order P X we asso

Pred z ciate the lower set Lx Pred x and the upper set Ux

P P

zSucc x P

CHAPTER INTERVAL DIMENSION AND DIMENSION

the case x MaxP is settled by the convention Ux X

Dene BP fLx Ux x Xg ordered by setinclusion

Note that this construction is in fact equivalent with Cogis construction in the

context of Ferrers dimension Co Cogis also uses Lx but replaces Ux by

the equivalent set fz X Succ x Succ zg He also proves Theorem but

in a dierent way and without the geometrical interpretation that our approach

is based on

The preceding lemmas prove that l Lx and u Ux together form

x x

an order preserving mapping from BI to BP hence

idimP dim BI dim BP

To get more structure into interval realizers we now intro duce a pro cedure

that transforms an interval extension I f a b x P g of P into its nor

x x

malization I f a b x P g

x x

In the rst step of the normalization we up date left endp oints

a maxfb z Predxg if x P n MinP

a minfa z MinP g otherwise

In the second step we up date right endp oints

z Succxg if x P n MaxP minfa b

z x

b maxfb z MaxP g otherwise

Note that the interval order I need not b e isomorphic to I In general I

is a sub order of I and a minimal interval extension see HMR for minimal

interval extensions of P if all the a b were dierent

x x

If P is realized by I fI I g then I fI I g realizes P as well

We call the b ox emb edding corresp onding to I the normalized b ox emb edding

of I For an example see Figure

After normalizing we have a realizer I fI I g of P interval represen

j j

tations a b and an asso ciated p oset of extreme corners BI fl u

x x

x P g The next theorem shows that the geometrically dened BI and the

combinatorially dened BP are isomorphic

Theorem If I is a normalized realizer of P then BI BP

REDUCING INTERVAL DIMENSION TO DIMENSION

r r r

r r

r P

r r

r r r

Figure P an interval realizer of P and its normalization

Pro of First observe that b oth partial orders have a least element generated by

x MinP as l and Lx resp ectively and a greatest element generated by

x MaxP as u and Ux resp ectively

Moreover l Lx and u Ux denes an order preserving mapping by

x x

the ab ove remarks To show the converse we distinguish four cases

is an interval Ux Ly We know that x Ux so x Pred y Since I

j j

l extension of P we obtain b a for all j Hence u

x y

y x

j j j

Lx Ly Rememb er that a maxfb z Predxg and a

x z y

j j j

maxfb z Pred yg By assumption Predx Pred y so a a and

z x y

l l

x y

Ux Uy By Lemma this is equivalent to Succx Succ y Now

u u follows symmetrically to the second case

x y

Lx Uy Since I is normalized there are z Predx and z Succ y

j j j j j

with a b and b a The hyp othesis provides z z hence a

x z y z x

j j j j

b b a b The validity of this inequality for all j again gives

z z z y

l u 2

x y

Now we are going to prove our main theorem ab out interval dimension and

dimension

Theorem dim BP idimP

Pro of The inequality dim BP idimP was a simple consequence of the def

inition of BP For the converse we need two arguments We rst show that a

CHAPTER INTERVAL DIMENSION AND DIMENSION

linear extension L of BP induces an interval extension I of P Secondly we

prove that if L L is a realizer of BP then the induced interval extensions

I I form an interval realizer of P

j L

Let L M M M b e a linear extension of BP For each x P there

are i j f rg such that M Lx and M Ux From Lx Ux and

i j

x Lx x Ux we obtain that i j So we can asso ciate with x the unique

interval a b i j We now show that the interval order I induced by

x x

the interval representation f a b x P g is an extension of P If x y in P

x x

then Ux Ly and thus with M Ux and M Ly b i j a

x y

i j

which implies x y in I

Let fL L g b e a realizer for BP The induced family of interval exten

sions fI I g of P is an interval realizer i all incomparabilities xjjy of P are

realized If xjjy in P then Ux Ly since x Ux but x Ly Therefore

j j

Ly precedes Ux in some L which gives a b The symmetric argument

y x

i i

yields an i with a b Both inequalities together give xjjy in I 2

x y

This theorem together with Inequality shows that dim BI is indep endent

of the interval realizer I

Consequences

In the previous section we intro duced the op eration P BP mapping partial

orders to partial orders We will now investigate several connections to other

ordertheoretical topics and results Since BP always has a greatest and a least

element we adopt the convention to call orders with this prop erty closed and

denote by Q the closure of any partial order Q ie the order resulting from Q

by adjoining a new greatest and a new least element

We rst lo ok at the eect of the op erator B applied to sp ecial classes of orders

Crowns and the Standard Examples

Let S denote the standard p oset of dimension n ie the set of all subsets of an

n element set with either or n elements ordered by setinclusion Then

BS S

n n

CONSEQUENCES

Let C denote the rcycle The rcycle is the dimensional p oset on r

elements fx y x y x y g with comparabilities

r r

x y y x x y x y y x

r r r

BC C

r r

Let the diagram of T b e a tree The truncation of T denoted by trT is the

induced tree on the nonleaf vertices of T Then

BT trT

In particular shows that the standard example S of a ndimensional order

is up to closures a xed p oint of the op eration P BP thus showing again

that for every n there is a p oset P with dim P idimP n

B and the Split Op eration

We now turn to the natural question whether for every closed order Q there is

some P with Q BP The next theorem answers this question armatively

Moreover it turns out that the op eration P BP is an almost left inverse of

the split op eration S which has applications in dierent branches of p oset theory

see eg Tr Fr Fe The split SP of an order P is the p oset of height one with

minimal elements fx x P g maximal elements fx x P g and ordered pairs

x y i x y in P

Theorem BSP P

Pro of For x P let Predx Pred x fxg Obviously P is isomorphic to

the setsystem fPredx x P g ordered by inclusion We will show that the sets

of BSP are just the primed sets Predx ie Predx fx x Predxg

together with the greatest element fx x x P g and the least element We

have

Lx and

Pred z Since x Succ x Ux Pred x Predz Ux

00 0

zSucc x z Succx

On the other hand Predx Pred z for z Succ x Together this gives

Ux Predx Similarly we obtain

Lx Predx Pred x Finally

Ux fx x x P g by denition since Succ x 2

CHAPTER INTERVAL DIMENSION AND DIMENSION

It is easy to verify that

BP BP

So we may generalize the theorem to

g n closures

n n

P B S P

Investigations on the eect of iterated splitting to the dimension TM lead to

the inequality

dim P dim S P dim P for all n

As a consequence of and we obtain that for every n there is an order

P such that

dim P dim B P for all k n

Just take P S Q for some order Q If we cho ose Q however to b e an m

dimensional interval order we obtain a large dierence in dimension with the next

iteration ie

dim P dim B P m

The Interval Dimension of Sub divisions

With the next theorem we relate the interval dimension of sub divisions of P to

the dimension of P Spinrad Sp showed that the dimension of a sub division

of a partial order can b e an arbitrary multiple of its dimension thus answering

Trotters Problem in Tr With our result we establish a theoretical framework

for his examples

In this context partial orders and their diagrams are regarded as directed

graphs whose edges x y corresp ond to ordered pairs and cover pairs x y

of P resp ectively An edge x y is subdivided by placing a new vertex z in

the middle of the edge ie x y is replaced by x z and z y In the case

of partial orders we then have to ensure transitivity ie all edges a z with

a Pred x and z b with b Succ y are also added

The complete diagram subdivision DS P is the sub division of all edges of

the diagram of P The complete subdivision CSP is the sub division of all the

CONSEQUENCES

edges of P ie of the transitively closed relation Since P is an induced sub order

of each of its sub divisions SubP and since SubP is an induced sub order of

CSP we obtain

dimP dim Sub P dim CSP

With the next theorem we give an upp er b ound for idim SubP

Theorem idim Sub P idim DSP idim CSP dim P

Pro of Take any emb edding of P into IR with k dimP and grow the p oints

to obtain an emb edding by minib oxes An interval emb edding of a sub division

SubP is then obtained by adding the b ox with lower extreme corner u and

upp er extreme corner l for the p oint z sub dividing the edge x y see Figure

This gives idim Sub P dimP for every sub division SubP

r r

r r r

r r r r

r r

Figure P a minib ox emb edding of P and the b ox emb edding of DSP

To prove that idim DSP idim CSP dimP we will show that P can b e

emb edded in BDS P This will give dimP dim BDS P From Theorem

we know idim DSP dim BDSP so we obtain dim P idim DSP

To show that P can b e emb edded in BDS P we apply the normalizing pro

cedure to the b ox emb edding I of DSP constructed in the rst paragraph of

the pro of During the normalization we can only shift the left endp oints of

intervals corresp onding to elements in MinP and the right endp oints of ele

ments in MaxP We then emb ed P into the lower extreme corners of the mini

b oxes of the normalized representation I This gives dimP k idimI

idimDSP 2

CHAPTER INTERVAL DIMENSION AND DIMENSION

Note that we obtained in fact a slightly stronger result If CSP SubP

DSP then BSub P VS P where VS P denotes the vertical split of P ie

the order obtained from P by substituting each vertex by a chain In TM a

distinct pro of for dimP idim VSP has b een given

Comparability Invariance of Interval Dimension

For the denition and basic facts on comparability invariance see Ha Let

CompP b e the comparability graph of P We will show that CompBP is

a comparability invariant of P in the sense that if CompP Comp Q then

CompBP CompBQ Together with Theorem and the known fact

that dimension is a comparability invariant this gives an alternative pro of of the

comparability invariance of interval dimension in the nite case The compara

bility invariance of interval dimension was rst shown in HKM

Theorem CompBP is a comparability invariant of P

Pro of Let A b e an autonomous subset of P It is enough see eg DPW to

A A

denotes the order resulting where P show that CompBP Comp BP

d d

A A

from substituting A by its dual A in P

Note rst that BA fLa Ua a Ag is a closed sub order of BP Let

BA b e BA without its greatest element and its least element Our

BA BA

claim is that BA is autonomous in BP To see this observe rst that for each

a A we can decomp ose Pred a into Pred a Pred A Pred a hence

the same is valid for all elements of BA On the other hand the elements of

BP n BA either contain all of A or their intersection with A is empty Now if

M BP n BA contains all of A then it also contains Pred A and M is ab ove

g g

all sets in BA If M Pred A then M is b elow all sets in BA In all the

other cases M is unrelated to all of BA This gives the claim

To settle the theorem we again need an analogue of namely BA

d k

BA Consider a normalized b ox emb edding of A in IR Its extreme corners

are an emb edding of BA into IR Flip the emb edding ie reverse the relations

this gives an emb edding of A and the extreme corners form an emb edding of

BA 2

As we have seen autonomous sets in P induce autonomous sets in BP The

converse however is far from b eing true Take as P a prime interval order then

REFERENCES FOR CHAPTER

j j

nontrivial autonomous BP is a chain Hence P has none but BP has

sets

Remarks on Computational Complexity

The transformation P BP obviously can b e computed in p olynomial time

thus this pap er also explains and reproves the earlier noticed coincidence in com

plexity for the recognition of partial orders of xed interval dimension Ya and

the related complexity status of recognizing trap ezoid graphs Ch MS HM In

particular it answers the questions of Dagan Golumbic and Pinter DGP ab out

the comparability invariance of interval dimension and the recognition of interval

dimension at most in a very direct and simple way

References for Chapter

BRT KB Bogart I Rabinovich and WT Trotter A b ound on

the dimension of interval orders Journal of Comb Theory A

Co O Cogis On the ferrers dimension of a digraph Discrete Math

Ch F Cheah A recognition algorithm for IIgraph Technical Rep ort

DGP I Dagan MC Golumbic and RY Pinter Trap ezoid graphs and

their coloring Discrete Applied Math

DM B Dushnik and E Miller Partially ordered sets Amer J Math

DPW B Dreesen W Poguntke and P Winkler Comparability invari

ance of the xed p oint prop erty Order

Fe S Felsner Orthogonal structures in directed graphs Preprint

Fi PC Fishburn Interval Orders and Interval Graphs Wiley NewYork

Fr A Frank On chain and antichain families of a partially ordered set

Journal of Comb Theory B

FHRT Z Furedi V Rodl P Hajnal and WT Trotter Interval

orders and shift graphs Preprint

CHAPTER INTERVAL DIMENSION AND DIMENSION

Ha M Habib Comparability invariants Annals Discrete Math

HKM M Habib D Kelly and RH Mohring Interval dimension is a

comparability invariant Discrete Math

HM M Habib and RH Mohring Recognition of partial orders with inter

val dimension two via transitive orientation with side constraints Preprint

HMR M Habib M Morvan and JX Rampon Ab out minimal interval

extensions Preprint

MS T Ma Algorithms on sp ecial classes of graphs and partial orders Ph

D Thesis Vanderbilt Univ

Or O Ore Theory of Graphs Amer Math So c Collo q Publ Vol

Ra I Rabinovich A note on the dimension of interval orders Journal of

Comb Theory A

Sp J Spinrad Edge sub division and dimension Order

TM WT Trotter and JI Moore The dimension of planar p osets

Journal of Comb Theory B

TM WT Trotter and JI Moore Characterization problems for

graphs partially ordered sets lattices and families of sets Discrete Math

Tr WT Trotter Stacks and splits of partially ordered sets Discrete

Math

Tr WT Trotter Problems and conjectures in combinatorial theory of

ordered sets Annals of Discrete Math

Ya M Yannakakis The complexity of the partial order dimension prob

lem SIAM J Alg Disc Meth

Chapter

Tolerance Graphs

Intro duction and Overview

An undirected graph G V E is called a tolerance graph if there exists a

collection I fI j x V g of closed intervals on the line and a tolerance

function t V IR satisfying

fx yg E jI I j mint t

x y x y

where jIj denotes the length of the interval I A tolerance graph is a bounded

tolerance graph if it admits a tolerance representation fI tg with jI j t for

x x

all x V

Tolerance graphs were intro duced by Golumbic and Monma GM We sum

marize two of the results proved there If all tolerances t equal the same value

c say then we obtain exactly the class of all interval graphs If the tolerances

are t jI j for all vertices x then we obtain exactly the class of all p ermutation

x x

graphs Furthermore the following theorem was proved

Theorem Every bounded tolerance graph is the complement of a com

parability graph ie a cocomparability graph

The most imp ortant article on tolerance graphs is due to Golumbic Monma

and Trotter GMT We summarize some of the results shown there in the next

theorem

CHAPTER TOLERANCE GRAPHS

Theorem

A tolerance graph must not contain a chord less cycle of length

greater than or equal to

A tolerance graph must not contain the complement of a chord less

cycle of length greater than or equal to

A tolerance graph admits an orientation such that every chord less

cycle is oriented as shown in Figure

c c

Figure Alternating orientation of C

Remark A graph G is called alternatingly orientable if there is an orientation

of G such that around every chordless cycle of length greater than the directions

of arcs alternate As a consequence of the preceding theorem we obtain that

tolerance graphs are alternatingly orientable see Br for more information on

this class of graphs

In this chapter we rep ort on some results which may b e useful for a complete

answer to the following op en problems

Problem Characterize tolerance and bounded tolerance graphs

Problem Is the intersection of tolerance graphs and cocomparability

graphs exactly the class of bounded tolerance graphs

Tolerance Graphs and Orders of Interval

Dimension

The starting p oint of this work is a representation theorem for b ounded toler

ance graphs Let G V E b e a b ounded tolerance graph with representation

fI tg and I a b We now dene interval subgraphs G G of G Let

x x x

TOLERANCE GRAPHS AND INTERVAL DIMENSION

G b e represented by the intervals I a t b and G by the intervals

x x x

I a b t It is easy to verify that G G G Since G and G are co

x x x

comparability graphs of interval orders P and P resp ectively the comparability

graph of P P P is the complement of G Therefore G is the co comparability

graph of an order with interval dimension at most A sp ecial feature of the

interval realizer fI I g of P is that jI j jI j for all x V In the spirit of

x x

the term b ox emb edding intro duced in Chapter we call such a representation

a square embedding The construction is illustrated in Figure

square

of x

a t

x x

Figure The square corresp onding to a b and tolerance t

x x x

Conversely let P X b e an order of interval dimension that admits

a square emb edding We claim that the co comparability graph of P CoCompP

is a b ounded tolerance graph Let fI I g constitute a square emb edding of

P and let the corresp onding intervals of x b e given by I a a l and

x x x

I a a l We now x some s IR such that s max a a The

x x x x x

co comparability graph of P is the b ounded tolerance graph given by the intervals

I a s a l and the tolerances t s a a

x x x

x x x x

We have thus shown the following strengthening of Theorem

Theorem A graph G is a bounded tolerance graph i G is the cocom

parability graph of an order P with interval dimension at most which

has a square representation

CHAPTER TOLERANCE GRAPHS

Co comparability graphs of orders with interval dimension at most are known

as trapezoid graphs An observation similar to ours is used by Bogart et al

BFIL to show that the class of b ounded tolerance graphs coincides with the

class of parallelogram graphs ie trap ezoid graphs where every trap ezoid is a

parallelogram

There exist orders of interval dimension which do not admit a square rep

resentation This is shown with the following example

Example It is easy to see that the graph G given in Figure is not alter

natingly orientable An orientation which is alternating on the cycles

and and contains would require

a contradiction Therefore G is not a tolerance graph

On the other hand G CoCompP and the order P has a b ox emb edding

which proves that idimP

k k

l k

k k

Figure The graph G CoComp P and the order P

Until now we only dealt with b ounded tolerance graphs As a consequence

of the next theorem we will obtain Every tolerance graph which is a co

comparability graph is a trapezoid graph This will b e useful later when we

characterize all tolerance graphs that are complements of trees and could b e of

use as well for a solution to Problem

TOLERANCE GRAPHS AND INTERVAL DIMENSION

Theorem The intersection of cocomparability graphs and alternat

ingly orientable graphs is contained in the class of trapezoid graphs

The main ingredient into the pro of of this theorem will b e a characterization

of orders of interval dimension Lemma which is due to Cogis Co also

Let P X and x y z t b e a in P We call the pairs x t and

z y the diagonals of the With P we now asso ciate the incompatibility

graph F

As vertices of F we take the pairs x y and y x whenever xjjy

Two vertices of F are connected by an edge i they are the diagonals of a

common in P

Lemma Cogis idimP i F is bipartite

Cogis obtained this result in the more general context of the Ferrersdimension

of directed graphs His denition of the incompatibility graph asso ciated with a

directed graph is somewhat more involved see HM pages for details In

the case of an antireexive and transitive digraph ie a partial order however

an easy case analysis shows that we have chosen the right denition

Pro of Theorem We have to show that idimP implies that CoCompP

is not alternatingly orientable

Let P with idim P b e given From Lemma we know that F contains

o dd cycles Fix an o dd cycle C x y x y x y x y

k k

in F If x y and x y are consecutive elements of C then by the

P i i i i

denition of F x y y x x is a cycle in CoComp P Therefore

P i i i i i

* An alternating orientation of CoCompP will either contain the two

arcs x y and y x or the two arcs y x and x

i i i i i i i

Assume that an alternating orientation A of CoCompP is given wlog we

may require x y to b e in A Using * we obtain that y x is in A Using

* again we obtain that x y is in A Rep eating this argument we nally

nd x y and hence y x in A This contradicts the existence of

k k

an alternating orientation 2

CHAPTER TOLERANCE GRAPHS

Some Examples

After having obtained the previous theorem I had the idea that the intersection of

co comparability graphs with alternatingly orientable graphs and the intersection

of co comparability graphs with tolerance graphs could b e the same In this part

we will separate several classes of graphs by examples

We rst need a denition Let P X b e an order of interval dimension

with a realizer ie a b ox emb edding I f a b x Xg I

x x

f a b x Xg We say x y X have crossing diagonals if the line

x x

segments a a b b and a a b b intersect in IR

x x x x y y y y

Lemma If an order P of interval dimension has a box embedding

without crossing diagonals then G CoComp P has an alternating orien

tation

Pro of We rst indicate how to dene an alternating orientation on G Using

the co ordinates of the b ox of x we dene two regions in the plane see Figure

R x f u v u a and v b and dist u v b a distu v a b g

x x x x x x

R x f u v u b and v a and distu v b a dist u v a b g

x x x x x x

Note that if xjjy then the diagonal of the b ox of y has to intersect either R x

a b

x x

Figure The regions dened by a b ox

or R x It can not intersect b oth since there are no crossing diagonals If the

diagonal of y intersects R x we orient the edge fx yg from x to y and if the

diagonal of y intersects R x we orient it from y to x

We now have to show that the orientation of G is alternating Since G is a

co comparability graph we only have to deal with cycles of length A C in G

TOLERANCE GRAPHS AND INTERVAL DIMENSION

corresp onds to a in P With this remark it is not hard to verify that our

orientation is alternating on the cycle 2

Consider the co comparability graph G of the order P given in Figure

From Lemma we know that G is alternatingly orientable but due to a cyclic

dep endence among the b oxes of a b c and d there is no square representation

of P Therefore G is not a b ounded tolerance graph But note that whenever

any b ox is deleted from the picture the representation can b e transformed into a

square representation Hence G is a minimal obstruction for the class of b ounded

tolerance graphs The example is quite stable with resp ect to this prop erty

Figure The complement of this order is alternatingly orientable but not a

b ounded tolerance graph

we may add an arbitrary subset of the following comparabilities to P without

aecting it

In Theorem and Theorem we will exhibit more examples containing

several innite families of obstructions for the class of b ounded tolerance graphs

It would b e quite hard to give a rigorous pro of that the graph G from the

previous example is no tolerance graph at all Our aim however is to exhibit

co comparability graphs which p ossess an alternating orientation but are not tol

erance graphs This can b e done using some more notation

A vertex x of G is called assertive if for every tolerance representation fI tg

of G replacing t by mint jI j leaves the tolerance graph unchanged An

x x x

assertive vertex never requires unb ounded tolerance Therefore if every vertex of

a tolerance graph G is assertive then G is a b ounded tolerance graph In GMT

the following sucient condition for a vertex to b e assertive is shown

CHAPTER TOLERANCE GRAPHS

Lemma Let x be a vertex in a tolerance graph G X E

If Adj x n Adjy for al l y with fx yg E then x is assertive

Lemma If G X E is not a bounded tolerance graph then G is not

a tolerance graph Here G is the graph on two copies X X of X with

edges fx x g for al l x X and fx y g for al l fx yg E That is we

i j

replace each vertex of G by a clique to obtain G

Pro of It is easy to check that every vertex in G meets the conditions of Lemma

ie is assertive Therefore if G is a tolerance graph it will also b e a b ounded

tolerance graph This however is imp ossible since even G is not a b ounded

tolerance graph 2

Now consider the order given in Figure The co comparability graph of

this order is just G if G is the graph corresp onding to Figure We have seen

that G is not a b ounded tolerance graph so G is not a tolerance graph but G

has an alternating orientation since again there are no crossing diagonals

Figure The complement of this order is alternatingly orientable but not a

tolerance graph

Cotrees and More Examples

Complements of trees are co comparability graphs In GMT it is suggested to

take them as an initial step towards a solution of Problem Using Theorems

and we obtain

TOLERANCE GRAPHS AND INTERVAL DIMENSION

alternatingly

orientable

tolerance

b ounded

tolerance

Example Fig

trap ezoid

co comparability

Figure Relations of tolerance graphs to other classes of graphs

Theorem If T is the complement of a tree T then the fol lowing con

ditions are equivalent

T is a tolerance graph

T is a bounded tolerance graph

T is a trapezoid graph

T is a tolerance graph

T contains no subtree isomorphic to the tree T of Figure

T is a tolerance graph then by Theorem T is the comparability Pro of If

T is a trap ezoid graph Orders graph of an order of interval dimension ie

which have trees as comparability graphs are of height one In a b ox realizer

of a height one order with interval dimension we can grow the b oxes of maxi

CHAPTER TOLERANCE GRAPHS

T T

c c c

Figure The trees T and T

mal elements upwards and b oxes of minimal elements downwards to make them

T is a tolerance graph then T is a b ounded tolerance graph squares Therefore if

If T is a b ounded tolerance graph then trivially it is a tolerance graph and

by Theorem it is a trap ezoid graph This gives the equivalence of and

For the equivalence of and we need a characterization of those trees

which are the comparability graphs of orders of interval dimension In chapter

we showed that for every p oset idimP dimBP If T is a tree then BT

is the truncation of T ie the tree obtained by cutting down the leaves of T

see page Among the irreducible orders of dimension there is only one tree

namely T From this we obtain that T is the unique tree among the obstructions

against interval dimension

For the remaining equivalence ie to show the equivalence of with we

refer to GMT 2

We now come back to minimal obstructions for the class of b ounded tolerance

graphs A quite simple observation will provide us with many examples

It is well known that a graph G is b oth a comparability graph and a co com

parability graph i G and G are comparability graphs of orders of dimension

An order P is called irreducible if dim P but whatever vertex x we remove

from P we obtain an order of dimension ie dimP for all x

Now let P b e irreducible and G CompP G is not a b ounded tolerance

graph since it is not a co comparability graph But if we remove any vertex x from

G then G will b e the co comparability graph of an order of dimension This

order has an emb edding by p oints hence a square emb edding by minisquares

Therefore G is a b ounded tolerance graph

Theorem If P is a irreducible order then the comparability graph of

P is a minimal obstruction for the class of bounded tolerance graphs

TOLERANCE GRAPHS AND INTERVAL DIMENSION

Remark A complete list of irreducible orders has indep endently b een com

piled by Kelly and by Trotter and Mo ore see KT They found isolated

examples and innite families

We now turn to a second large class of obstructions Recall that in the pro of of

Theorem we gave evidence that a height order of interval dimension admits

a square emb edding ie its co comparability graph is a b ounded tolerance graph

An order P is called interval irreducible if idimP but whatever vertex x

we remove from P we obtain an order of interval dimension ie idimP

Now let P b e a interval irreducible order of height and G CoComp P

From idimP we conclude that G is not a tolerance graph But if we remove

any vertex x from G then G is the co comparability graph of an order p osessing

a square emb edding Hence G is a b ounded tolerance graph

Theorem If P is a interval irreducible order of height then the

cocomparability graph of P is a minimal obstruction for both the class of

tolerance graphs and the class of bounded tolerance graphs

Remark A complete list of the interval irreducible orders of height has b een

compiled by Trotter Tr There are isolated examples and innite families

We close this chapter with a last example Let Nx Adjx fxg denote

the neighbourhood of a vertex x in G A set of vertices fx x x g is called an

asteroidal triple if any two of them are connected by a path which avoids the

neighb ourho o d of the remaining vertex In GMT it is shown that co compara

bility graphs do not contain asteroidal triples hence b ounded tolerance graphs

are asteroidal triplefree as well More information on asteroidal triplefree graphs

is given in COS

All examples of tolerance graphs which are not b ounded tolerance graphs

given in GMT are not asteroidal triplefree Therefore it seems plausible to

conjecture that every tolerance graph which is not b ounded contains an asteroidal

triple Using Theorem we now show that this is not true in general

Example Let G b e the comparability graph of the order H from the list of

irreducible orders see Figure This graph is a tolerance graph and asteroidal

triplefree but by Theorem it can not b e a b ounded tolerance graph

CHAPTER TOLERANCE GRAPHS

n n

A B D

n n

A D

n n

B C

t jI j

t t t t t

A C D

Figure G CompH and a tolerance representation of G

References for Chapter

BFIL KP Bogart PC Fishburn G Isaak and L Langley Prop er

and unit tolerance graphs DIMACS Technical Rep ort

Br A Brandstadt Sp ecial graph classes a survey Forschungsergebnisse

FSU Jena N

Co Cogis Dimension Ferrers des graphes orientes PhDthesis Paris

COS DG Coneil S Olariu and L Stewart On the linear structure

of graphs COST Workshop held at Rutgers Univ

DDF JP Doignon A Ducamp and JC Falmagne On realiz

able biorders and the biorder dimension of a relation J of Math Psych

FHM S Felsner M Habib and RH Mohring On the interplay of

interval dimension and dimension Preprint TUBerlin

GM MC Golumbic and CL Monma A generalization of interval

graphs with tolerances Congressus Numeratium

GMT MC Golumbic CL Monma and WT Trotter Tolerance

graphs Discrete Applied Mathematics

HM M Habib and RH Mohring Recognition of partial orders with

interval dimension two via transitive orientation with side constraints

Preprint TUBerlin

KT D Kelly and WT Trotter Dimension theory for ordered sets in

I Rival ed Ordered Sets D Reidel Publishing Company

Index

covering acyclic orientation

chromatic numb er

algorithm

closure

Faigle Schrader

co comparability graph

greedy

comparabilities

linear extension

comparability graph

list starty

comparability invariance

list Tgreedy

cover

starty

Tgreedy

defect

sequence

diagram

alternatingly orientable

Dilworths theorem

antichain

dimension

covering

directed graph

partition

distinguishing row

arc

down set

balanced

onedominant

endp oint

zerodominant

extension

arcchromatic numb er

Ferrersdimension

arccoloring

assertive

greedy

asteroidal triple

autonomous

Hamiltonian cycle

auxiliary list

Hamiltonian path

Hasse diagram

height

idim

Bo olean lattice

incidence matrix

b ounded tolerance graph

incompatibility graph

b ox emb edding

injection

bump

interval dimension

canonical interval order

interval extensions

canonical representation

interval graph

chain interval order

INDEX

canonical semitransitive

interval realizer shift graph

interval representation shiftgrap

irreducible shiftgraph

SMin

jump

Sp erners theorem

alpha

split

b eta

square emb edding

jump counters

staircase

jump numb er

length

standard p oset

lexicographic order

Start see starting element

line graph

starting elements

linear extension

step

algorithm

step graph

greedy

Sterlings formula

optimal

sub division

starty

complete

linear order

complete diagram

magnitude

Succ order

marking

successors

function

tiebreaking row

Lmarking

tolerance graph

maximal element

b ounded

minimal element

transition

minimal interval extension

counter

transitive tournament

normalization

trap ezoid graphs

NPcomplete

truncation

NPhard

width

parallelogram graphs

partial order

reexive

strict

Pred order

predecessors

rcycle

realizer

semiorder