Bounds to the page number of partially ordered sets - Springer Link

BOUNDS TO THE PAGE NUMBER OF PARTIALLY ORDERED SETS &

M a c i e j M. S y s l o @ Technische Universitaet Berlin, FB 3 - Mathematik Str. d e s i7. J u n i 13S, I 0 0 0 B e r l i n 12

ABSTRACT

In t h i s number,

paper

we initiate

well-known

study

for graphs.

then they are used to evaluate page number

I.

for some

families

of a new poset

Lower

and upper

and to bound

invariant,

bounds

the exact

of posets.

Several

(which

a

She p a g e

are derived value

problems

of

and the

a r e posed.

Introduction

A book consists pa6es. Thus,

Each

of a spine

paffe is a h a l f - p l a n e

any half-plane (vertical)

G=fV, E)

in a b o o k c o n s i s t s

I.

place assign

the vertices each edge

assigned Instead a circle.

of takin~ Then

chords

correspondin~

and

is a 2 - p a g e

The

number

as its

and a plane

book.

a

boundary.

with

embedding

of

a

distin-

of

a

~raph

o f t w o steps:

V on the spine

in s o m e

t o o n e of t h e p a g e s

order,

and

in s u c h a way,

that the edges

t o o n e p a ~ e d o n o t cross. a linear

the edges

S t e p 2 is t o c o l o r secting

line

line)

that has the spine

is a one-paffe b o o k

guished

2.

is

of vertices,

of the graph

the chords

fret t h e s a m e circle

order

~raph,

with

color see

become

few colors - this

[2].

It

we may place

chords

of the circle

so that

is e q u i v a l e n t is

them on

no

two

to

well-known,

& This research was partially supported by the grant the Institute of Informatics, University of Warsaw.

inter-

color

the

that

the

RP.I.09

@A Fellow of the Alexander yon Humboldt-Stiftung (Bonn). from Institute of Computer Science, University of Wroclaw, k i e ~ o 20, 5 i 1 5 1 W r o c l a w , P o l a n d .

and

from

On leave Przesmyc-

182

c i r c l e g r a p h s are e x a c t l y the o v e r l a p ~ r a p h s

[4].

~2 .... ,~ ) be an o r d e r of the v e r t i c e s of G

alon~

each e d g e (u,u}

is i n t e r p r e t e d as an o p e n

r i g h t ends are r e s p e c t i v e l y g e r s ~-Ifu) and ~-1(u). f a m i l y ~={luu: O(G,H)

Hence, the

let

H=(~i,

spine.

Now,

interval Iuu w h o s e left and

the s m a l l e r and the ~ r e a t e r of two

inte-

We c o n s t r u c t the o v e r l a p g r a p h OfG, H) of

{u,u}eE(G)}

the

and S t e p 2 a b o v e is now to c o l o r the graph

optimally.

The p~6e numSer pRfG)

of G is the s m a l l e s t n u m b e r &

such

that

G

has a b o o k e m b e d d i n g on & pa~es. The p r o b l e m of e m b e d d i n g ~ r a p h s in books was first B e r n h a r t and K a i n e n is m o t i v a t e d

[i].

The r e c e n t i n t e r e s t

by t h e i r a p p l i c a t i o n s

p r o c e s s o r arrays,

see C h u n ~ et al°

In some a p p l i c a t i o n s m e n t i o n e d

i n v e s t i g a t e d by

in such ~raph e m b e d d i n g s

in VLSI d e s i g n

of

fault-tolerant

[2]. in [2],

u s i n ~ s t a c k s and q u e u e s in parallel,

fop

instance

in

sorting

the set of f e a s i b l e o r d e r i n g s of

v e r t i c e s on the s p i n e is r e s t r i c t e d to a c e r t a i n s u b f a m i l y of

permu-

tations.

exten-

If this f a m i l y c o i n c i d e s with the set of all

linear

sions of a p a r t i a l l y o r d e r e d set t h e n the book e m b e d d i n g p r o b l e m g e n e r a l ffraphs r e d u c e s to that for (Hasse) of o r d e r e d sets,

see

for

diagrams - coverin~ graphs

[P].

Let P=(P,~) be a p a r t i a l l y o r d e r e d set ( s i m p l y c a l l e d a pose( d e n o t e d by P) and let ~(P) d e n o t e the set of all

and

l i n e a r e x t e n s i o n s of

P (a li~eee e x f e n s i o n of F is a total o r d e r of P w h i c h p r e s e r v e s r e l a t i o n ~).

The Hasse

dia6ram H(P) of P is t h e

~raph

whose

set c o r r e s p o n d s to P and tWO v e r t i c e s u , u ~ P are a d j a c e n t

is a c o n n e c t e d ~raph.

In w h a t f o l l o w s we

vertex

HfP)

in

and only if e i t h e r u c o v e r s u or u c o v e r s u in P. A p o s e r P R e e f e d if HfP)

the

if

is

tom-

assume

that

all p o s e r s are connected.

A boo~ e m ~ e d d i n 6 of e pose( P ~ i ~ h respeef

fo beZCP)

is

the

em-

beddinff of H(F) w i t h its v e r t i c e s (i.e~ ; e l e m e n t s of P) p l a c e d on the s p i n e in a c c o r d a n c e with L. Let O(P,i) d e n o t e the c o r r e s p o n d i n g o v e r lap graph.

The page n u ~ e r

pnfP, L) o~ P ~ifh

s m a l l e s t n u m b e r ~ of p a g e s s u c h that HfP) pa~es~

or e q u i v a l e n t l y

respeef

fo

i

has a b o o k e m b e d d i n ~

is

the on

k

183

w h e r e x d e n o t e s the c h r o m a t i c The page n u m b e r

pn(P)

pulP)

n u m b e r of a graph.

of P is d e f i n e d as follows:

= min(pn(P,L):

L~Z(P)}.

T h e r e e x i s t c o m p l e t e c h a r a c t e r i z a t i o n s of g r a p h s with p a g e i and 2 [i].

It is a l s o k n o w n that p n f G ) ~ 4 for e v e r y p l a n a r

and t h i s b o u n d

is t i g h t (see Y a n n a k a k i s

p o i n t of view,

recognizing

as d i f f i c u l t as f i n d i n g cycle, ver,

[12]).

if a (planar)

From

the

a

w h i c h is an N P - c o m p i e t e p r o b l e m (see W i g d e r s o n

of the p r o b l e m i n s t a n c e is a l s o N P - c o m p l e t e ,

of posers°

G

Hamiltonian

[ii]).

o p t i m a l b o o k e m b e d d i n g of G w h e n t h e v e r t e x o r d e r i n g

The p u r p o s e of t h i s p a p e r

graph

algorithmic

~ r a p h G has p a g e n u m b e r 2 is

if a m a x i m a l p l a n a r g r a p h has

lem for c i r c l e g r a p h s is N P - c o m p l e t e ,

number

see

is

Moreoa

part

s i n c e the c o l o r i n g p r o b -

[33.

is to i n i t i a t e s t u d y of b o o k

embeddings

We p r o v i d e for p o s e r s c o u n t e r p a r t s of s o m e of the

results

o b t a i n e d for g r a p h s and p r e s e n t some new ideas and r e s u l t s w h i c h hold o n l y for posers.

The m o s t

i m p o r t a n t o n e s are o b t a i n e d by u s i n g

lower

and u p p e r b o u n d s to the p a g e number. W h e n t h i s p a p e r was c o m p l e t e d ,

the a u t h o r r e c e i v e d a c o p y of

w h i c h a l s o c o n t a i n s s o m e b o u n d s to the p a g e n u m b e r of posers,

[iO]

however

in t e r m s of o t h e r invariants.

Z.

Lower

and

upper

Let P=(P,~)

bounds

be a p o s e r with n e l e m e n t s and m e d g e s

linear e x t e n s i o n LEa(P)

can be e x p r e s s e d

H(P).

in

Each

as a d i r e c t sum of c h a i n s of

P,i.e.

L

=

CO +

CI

+

..o

+

C~,

w h e r e C i (O~iS~)

is a c h a i n of P. The n u m b e r

comparabilities)

between consecutive elements

(2)

of

breaks

in

L

is

s(P,L) and c a l l e d the j u m p n u m b e ~ of F w i t h respect

to

n u m b e r s ( P ) of P is equal to m i n i m u m of sfP, b) a n d

the

b(P) of P is equal to m a x i m u m of s(P,L), L in ~(P).

The p r o b l e m of f i n d i n g s(P)

tite posets, time (see

non-

denoted L,

The

bump

by ju~

number

in b o t h c a s e s t a k e n o v e r all

is N P - c o m p l e t e e v e n for b i p a r -

w h e r e a s the b u m p n u m b e r can be c a l c u l a t e d

[6]).

(i.e.

in

polynomial

184

Let us consider the spine

a book embedding

according

P in L is c a l l e d

t o L.

and C3 (i of H(P), respec* clear

bound

to pn(P)

corresponds

where

that

LEFaIAZ.

e2=(x,~) if ~ v

by some

eletwo (u,~)

In ~ e n -

spine

edges

if L c o n s i s t s

of P accordin~ The

number

to

of

(3)

P satis£ies (4)

] + i g pn(P).

to a special

from

El).

matching

a sequen~uZ

of P accordin~

pa~e,

see also

of a poser

the clique

number

Although

the best

lower

as in Lemma

I (since

we intend

the pa~e number

Note

[23,

that

bound

for a linear

an

H(F,L)={el,e2, mu~c'h~n 6 in L.

~C~h It is

t o L, n o t w o e d g e s

of

Hence,

P with respect

t o LE~fP)

satisfies

(5)

o f a ~raph. of type

(3)

to minimize

extension l(a),

with

instance,

for a poser

F in Fig.

Fi~.

and bfP)=S,

but one can easily

ifb)

L

spine

~[O(P,L)) ~ pn(P,L) where o denotes

Ci

be

then

we have xlKx2K...Kx/