Graph Drawing �

Graph Drawing

Pro ceedings of the ALCOM International Workshop on Graph Drawing

Sevres Parc of Saint Cloud Paris

Septemb er

Edited by

G Di Battista P Eades H de Fraysseix

P Rosenstiehl and R Tamassia

DRAFT do not distribute

Septemb er

ALCOM International Workshop on Graph Drawing

Sevres Parc of Saint Cloud Paris

Septemb er

Graph drawing addresses the problem of constructing geometric representations of abstract

graphs and networks It is an emerging area of research that combines avors of graph theory

and computational geometry The automatic generation of drawings of graphs has imp ortant

applications in key computer technologies such as software enginering database design and

visual interfaces Further challenging applications can b e found in architectural design circuit

schematics and pro ject management Research on graph drawing has b een esp ecially active

in the last decade Recent progress in computational geometry top ological graph theory and

order theory has considerably aected the evolution of this eld and has widened the range of

issues b eing investigated

This rst international workshop on graph drawing covers ma jor trends in the area Pap ers

describ e theorems algorithms graph drawing systems mathematical and exp erimental analyses

practical exp erience and a wide variety of op en problems Authors come from diverse academic

cultures from graph theory computational geometry and software engineering

Supp ort from ALCOM I I ESPRIT BA and by EHESS is gratefully acknowledged we

would also like to thank Mr David Montgomery for preparing this do cument

Giusepp e Di Battista Univ Rome Italy dibattistaiasirmcnrit

Peter Eades Univ Newcastle Australia eadescsnewcastleeduau

Hub ert de Fraysseix CNRS France prosenstdmiensfr

Pierre Rosenstiehl EHESS France prosenstdmiensfr

Rob erto Tamassia Brown Univ USA rtcsbrownedu

Contents

J Pach New Developments in Geometric Graph Theory

Prosenjit Bose William Lenhart and Giusepp e Liotta Characterizing Proximity Trees

Franz J Brandenburg and Peter Eades A Note on a Separator Algorithm for Tree

Embeddings and its Application to Free Tree Drawings

Brian Regan Two Algorithms for Drawing Trees in Three Dimensions

Go os Kant Giusepp e Liotta Rob erto Tamassia and Ioannis G Tollis Area Require

ment of Visibility Representations of Trees

Ashim Garg and Rob erto Tamassia Ecient Computation of Planar StraightLine

Upward Drawings

Uli Fomeier and Michael Kaufmann An Approach for BendMinimal Upward Drawing

CThomassen Representations of Planar Graphs

Patrice Ossona de Mendez On Lattice Structures Induced by Orientations

Jan Krato chvl and Jir Matousek Complexity of Intersection Classes of Graphs

Hub ert de Fraysseix Patrice Ossona de Mendez and Pierre Rosenstiehl On Triangle

Contact Graphs

Simone Pimont and Michel Terrenoire Characterisation and Construction of the Rect

angular Dual of a Graph

Go os Kant and Xin He Two Algorithms for Finding Rectangular Duals of Planar Graphs

Go os Kant A More Compact Visibility Representation

Anna Lubiw Cone Visibility Graphs

Bo jan Mohar Circle Packing Representations in Polynomial Time

Bo jan Mohar Martin Juvan and Joze Marincek Obstructions for Embedding Extension

Problems

Antoine Bergey Automorphisms and Genus on Generalised Maps

Ivan Rival Upward Drawing on Surfaces

Bo jan Mohar and Pierre Rosenstiehl Tessalation and Visibility Representations of

Maps on the Torus

Teresa M Przytycka and Jozef H Przytycki A Simple Construction of High Repre

sentativity Triangulations

Prosenjit Bose Hazel Everett Sandor Fekete Anna Lubiw Henk Meijer Kathleen

Romanik Tom Shermer and Sue Whitesides On a Visibility Representation for

Graphs in Three Dimensions

Jianer Chen Saro ja P Kanchi and Jonathan L Gross On Graph Drawings with

Smal lest Number of Faces

Marc Bousset A Flow Model of Low Complexity for Twisting a Layout

Giusepp e Di Battista Giusepp e Liotta and Francesco Vargiu Convex and nonConvex

Cost Functions of Orthogonal Representations

Giusepp e Di Battista and Luca Vismara Topology and Geometry of Planar Triangular

Graphs

Frank Dehne Hristo Djidjev and JorgRudiger Sack An Optimal PRAM Algorithms

for Planar Convex Embedding

Miki Shimabara Miyauchi Algorithms for Embedding Graphs Into a page Book

Hossam ElGindy Michael Houle Bill Lenhart Mirka Miller David Rappap ort and

Sue Whitesides Dominance Drawings of Bipartite Graphs

Ulrich Finke and Klaus Hinrichs Computing the Overlay of Regular Planar Subdivi

sions in Linear Time

Alain Denise Generation of Random Planar Maps

Joseph Manning Symmetric Drawings of Graphs

Tomaz Pisanski Recognizing Symmetric Graphs

Peter Eades and Tao Lin Algorithmic and Declarative Approaches to Aesthetic Layout

Isab el F Cruz Rob erto Tamassia and Pascal Van Hentenryck A Visual Approach to

Graph Drawing

Gaby Zinmeister Layout of Trees with Attribute Graph Grammars

Sandra P Foubister and Colin Runciman The Display Browsing and Filtering of

Graphtrees

Daniel Tunkelang ALayout Algorithm for Undirected Graphs

Stephen C North Drawing Ranked Digraphs with Recursive Clusters

Ioannis G Tollis and Chunliang Xia Graph Drawing Algorithms for the Design and

Analysis of Telecommunication Networks

Michael Himsolt A View to Graph Drawing Algorithms through GraphEd

C L McCreary C L Combs D H Gill and J V Warren An Automated Graph

Drawing System Using Graph Decomposition

Michael Junger and Petra Mutzel Maximum Planar Subgraphs and Nice Embeddings

Practical Layout Tools

Peter Eades and Xavier Mendonca Heuristics for Planarization by Vertex Splitting

Takao Ozawa Planar Graph Embedding with a Specied Set of FaceIndependent Vertices

Bjorn Sigurd Benestad Johansen Implementation of the Planarity Testing Algorithm

by Demoucron Malgrange and Pertuiset

Jo el Small A Unied Approach to Testing Embedding and Drawing Planar Graphs

Hermann StammWilbrandt A Simple LinearTime Algorithm for Embedding Maxi

mal Planar Graphs

H de Fraysseix and P Rosenstiehl The LeftRight Algorithm for Planarity Testing

and Embedding

P O de Mendez Connexity of Bipolar Orientations

New Developments in Geometric Graph Theory

J Pach

Abstract not Available

Characterizing Proximity Trees

y z

Prosenjit Bose William Lenhart and Giusepp e Liotta

Much attention has b een given over the past several years to developing algorithms for

emb edding abstract graphs in the plane such that the resulting drawing has certain geomet

ric prop erties For example those graphs which admit planar drawings have b een completely

characterized and ecient algorithms for pro ducing planar drawings of these graphs have b een

designed For an overview of graph drawing problems and algorithms the reader is

referred to the excellent bibliography of Di Battista Eades Tamassia and Tollis More

over many problems in pattern recognition and classication geographic variation analysis

geographic information systems computational geometry computational morphology and com

puter vision use the underlying structure present in a set of data p oints revealed by means of a

proximity graph A proximity graph attempts to exhibit the relation b etween p oints in a p oint

set Two p oints are joined by an edge if they are deemed close by some proximity measure

It is the measure that determines the typ e of graph that results Many dierent measures of

proximity have b een dened The relatively closest graph the relative neighb orho o d graph

the gabriel graph the mo died gabriel graph and the delaunay triangulation are but

a few of the graphs that arise through dierent proximity measures

An extensive survey on the current research in proximity graphs can b e found in Jaromczyk

and Toussaint As the survey suggests interest in proximity graphs has b een increasing

rapidly in the last few years but most of the interest has b een algorithmic and little attention

has b een given to the combinatorial characteristics of these graphs Monma and Suri show

that any tree with maximum vertex degree ve can b e drawn as a minimum spanning tree

We study the problem of drawing trees as certain typ es of proximity graphs We say that a

tree T can b e drawn as a proximity graph when there exists a set of p oints in the plane such

that the proximity graph of that set of p oints is isomorphic to T Some sucient conditions

for such drawings to exist have b een given by Cimikowski for relatively closest graphs by

Matula and Sokal for gabriel graphs and by Urquhart for relative neighb orho o d graphs

However there exists no complete combinatorial characterization for any of the these typ es of

proximity graphs To this end we give a complete characterization of the trees that can b e

realized as either the relative neighb orho o d graph relatively closest graph gabriel graph or

mo died gabriel graph of a set of p oints in the plane

Research supp orted in part by NSERC and FCAR Scho ol of Computer Science McGill University

University Montreal Queb ec HA A jitmucsmcgillca

Department of Computer Science Williams College Willia ms town MA This work has b een done

when this author was visiting the Scho ol of Computer Science of McGill University lenhartcswillia msedu

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza via Salaria I Roma

Italia This work has b een done when this author was visiting the Scho ol of Computer Science of McGill University

liottainfokiting uni romai t

The suciency of the conditions of our theorems is established by providing drawing al

gorithms Given an abstract tree that admits a particular typ e of drawing say as a gabriel

graph the drawing algorithm will compute a set of p oints such that the gabriel graph of the

set of p oints is a tree that is isomorphic to the given tree As for the necesssity for each typ e

of proximity graph considered we exhibit a set of trees called forbidden trees which we show

cannot b e drawn as that typ e of proximity graph We then prove that no tree which can b e

drawn as such a proximity graph can contain any of the forbidden trees

Theorem A tree T can be drawn as the relatively closest graph or relative neighborhood graph

if and only if the maximum vertex degree of T is at most The drawing can be obtained in

O n time where n is the number of vertices in T in the real RAM model

Theorem A tree T can be drawn as the modied gabriel graph if and only if the maximum

vertex degree of T is at most three This drawing can be obtained in O n time where n is the

number of nodes in T in the real RAM model

The characterization of trees which can b e drawn as gabriel graphs dep ends on a set of

subgraphs called wide trees A rooted tree T u consists of a tree T and a distinguished vertex

u of T called the root of T Given a tree T a vertex v of T and a neighb oring vertex x of v

T v is dened to b e the comp onent of T v containing x

Denition A ro oted tree T u is wide if

deg u or

deg u u has neighb or v of degree four and for each neighb or x of v x u T v x

is wide

Denition A tree T is forbidden if it contains any of the following

A vertex of degree at least ve

Two adjacent vertices of degree four

A vertex v of degree four such that for each neighb or x of v the subtree T v x is wide

Theorem A tree T can be drawn as the gabriel graph if and only if T is not forbidden This

drawing can be obtained in O n time where n is the number of nodes in T in the real RAM

model

References

R Cimikowski Prop erties of some Euclidean proximity graphs Pattern Recognition Letters

G Di Battista P Eades R Tamassia and I G Tollis Algorithms for Automatic Graph

Drawing An Annotated Bibliography Dept of Computer Science Brown Univ Technical

Rep ort

K R Gabriel and R R Sokal A New Statistical Approach to Geographical Analysis Sys

tematic Zoology pp

J Hop croft and RE Tarjan Ecient Planarity Testing J ACM pp

JW Jaromczyk and GT Toussaint Relative Neighb orho o d Graphs and Their Relatives

Proceedings of the IEEE pp

P M Lankford Regionalization Theory and Alternative Algorithms Geographical Analysis

DW Matula and RR Sokal Prop erties of Gabriel Graphs Relevant to Geographic Variation

Research and the Clustering of Points in the Plane Geographical Analysis pp

C Monma and S Suri Transitions in Geometric Minimum Spanning Trees Bell Communi

cations Research Technical Rep ort

T Nishizeki and N Chiba Planar Graphs Theory and Algorithms Annals of Discrete

Mathematics North Holland

GT Toussaint The relative neighb orho o d graph of a nite planar set Pattern Recognition

RB Urquhart Some prop erties of the planar Euclidean relative neighb ourho o d graph

Pattern Recognition Letters pp

A Note on a Separator Algorithm for Tree Emb eddings and its

Application to Free Tree Drawings

Franz J Brandenburg and Peter Eades

Divide and Conquer is one of the ma jor principles for algorithm design It leads to fast

algorithms with go o d outputs particularly if the division partitions a problem instance into

two parts of almost equal size For a problem instance of size n the partition tree of solvable

subproblems is then a complete binary tree of size n This means only a linear growth in

size or an O expansion However if there is no strict partition into equal sized subproblems

then the partition tree may b e any binary tree Binary trees are separable ie they admit

a partition into two subtrees in the range In fact for every k n where n is the

size of the tree there is a subtree of size m and k m k Moreover there is a fast

search algorithm for a prop er subtree in the range searching in the bigger subtree until

its size falls under k There may b e a subtree t which approximates k much b etter than

the guaranteed range of k k and this t may lie somewhere in a smaller branch These

wellknown strategies are used in our emb edding algorithm

Algorithm DFSEMBED

DFSEMBED maps a binary tree t into a complete binary tree t such that each no de and each

edge of t are visited twice according to a depthrst traversal Moreover the ro ot of t is pinned

to the ro ot of t t should b e big enough It is easily reduced to minimal size

Let S v v v b e the sequence of no des of t visited in a dfs traversal by the recursion

ro ot left right ro ot Let P x x x b e the path in t from the ro ot to some designated

Universitat Passau Lehrstuhl fur Informatik Passau Germany brandenbinformatikun ip assau de

University of Newcastle Department of Computer Science Callahan NSW Australia

eadescsnewcastleeduau

no de x selected by a go o d strategy For i q let y b e the other child of x and let t

q i i i

denote the subtree of t with ro ot y Let t b e the subtree with ro ot x Rearrange P to

i q q

the sequence Q x x x x and consider the sequence of subtrees t t t t

q q q q

Rename x into x and t into t such that Q x x

q q q

For i q if t is nonempty then map its ro ot y to the next vertex v in the

i i

j i

sequence S such that the vertex v was unmarked and by recursion t ts into the complete

j i

subtree of t with ro ot v All vertices of this subtree will b e doubly marked Map the ancestor

j i

x of y to the ancestor w of v increase the mark of w and up date all ancestors of w by a

i i

j i

single markThe edge from x to y is directly mapp ed into the tree edge of t If t is empty

i i i

then map x to the next not yet doubly marked vertex in the sequence S and increase the mark

by one It remains to route the edges on the path x x x and from x to the second

child x For i q the edge from x to x is mapp ed to the path according to

q i i

the df s traversal of t with the edge from x to x along the rst traversal and the others along

the second The edge from x to x is routed along the second traversal from x to x

q q

For a illustration of DFSEMBED draw x x as the rightmost path in the left subtree

with each t b elow x and let t b e the right subtree of the ro ot x

i i

Algorithm DFSEMBED has the following prop erties

Lemma DFSEMBED runs in linear time

Let t b e a binary tree of size n If DFSEMBED cho oses the path P x x x Lemma

such that

i for i q the subtree with ro ot x is bigger than the other subtree t or

i i

ii siz et siz et siz et siz et or

iii i and ii hold

then DFSEMBEDt is a complete binary tree with height at most log n

Pro of Sketch DFSEMBED maps a sequence of chained trees T x t x t

q q

where x is the ancestor of t and x into a complete tree whose height exceeds the height of

i i i

an almost optimal binary search tree at most by one The access distribution for the search tree

log n log n

i i

is W for each x and W for each t of size n with W

i i i

We wo o d like to have an additive increase of the height only but for DFSEMBED the factor

c is b est p ossible

Lemma There is an innite sequence of binary trees T such that DFSEMBED T

i i

pro duces complete binary trees of height log n with n siz eT

i i i

Pro of Sketch Let the left subtree of T consist of a new vertex with two copies of T

i i

as its subtrees and let the right subtree of T b e trivial consisting eg of two vertices Let T

b e the complete tree with three vertices Then T has vertices By induction one of

the two T needs height at least log siz eT and DFSEMBED sets the ro ot of this

i i

T at depth

Corollary Every binary tree t of size n can b e emb edded into a complete binary tree t

of height c log n with c congestion and dilation O log n Thus the expansion is O n

each edge of t is used at most twice and each edge of t is mapp ed into a path of length at most

c log n

Remark The linear expansion obtained from the factor c for the increased height is

b etter than the one obtained by Hong et al HMR and Ruzzo and Snyder RS They demand

b ounded dilation and then achieve c Ruzzo p ersonal communication which yields an

expansion of O n

For free tree drawings or equivalently planar emb eddings of binary trees one may use DFS

EMBED as an intermediate step but this is a detour

For every binary tree t of size n there is a free tree drawing dt derived from Corollary

tree drawings of complete binary trees such that dt uses O n area and has edges with

O log n b ends

This result is noncomp etitive Valiant Va has shown that there are planar tree emb eddings

with O n area and O log n b ends and Crescenci et al CBP have upwards drawings with

O n log n area and no b ends

References

CBP P Crescenci G Di Battista and A Pip erno A note on optimal area algorithms for

upward drawings of binary trees Computational Geometry Theory and Applications

BET G Di Battista P Eades and R Tamassia Algorithms for Drawing Graphs An Anno

tated Bibliography

HMR JW Hong K Mehlhorn A L Rosenb erg Cost Tradeos in Graph Embeddings

with Applications Journal of the Asso c for Comput Machinery Vol No pp

Me K Mehlhorn Data Structures and Algorithms EATCS Monograph Computer Science

RS WL Ruzzo and L Snyder Minimum Edge Length Planar Embeddings of Trees In HT

Kung B Sproull and G Steele ed VLSI Systems and Computations pp

Va LG Valiant Universality Considerations in VLSI Circuits IEEE Transactions on Com

puters C pp

Two Algorithms for Drawing Trees in Three Dimensions

Brian Regan

This pap er explores the presentation of tree structures in three dimensions Some visualiza

tion systems have b egun to use three dimensional tree drawings however the wealth of funda

mental results on two dimensional tree layout for example

is not matched in three dimensions We are sp ecically interested in drawing free trees that is

trees with no presp ecied ro ot

Two extensions to existing tree drawing algorithms were develop ed The rst takes the H

tree drawing algorithm for binary trees and extends it into three dimensions This additional

dimension is achieved by alternating the direction of the next no de b etween three rather than

two axes The Htree algorithm pro duces a grid drawing The second algorithm takes the

principle of a radial drawing algorithm and extends it into three dimensions The successive

annuli of the radial algorithms are replaced by successive spherical shells sharing a common

Department of Management University of Newcastle New South Wales AUSTRALIA

centre Working from a ro ot no de at the centre for each no de an area for its descendants is

pro jected onto the next shell This area is compared with the area formed by the intersection

of the tangent plane at the given no de with the next outer shell This second area is the

equivalent of the annulus wedges formed in the two dimensional case The smaller of these two

areas is sub divided according to the numb er of children of the original no de The sub division

is accomplished by using a Lamb ert equal area pro jection of the sphere onto a plane and

allo cating rectangles for each of the child no des from which spherical co ordinates of the no des

are then determined Improvement to the nal layout will b e gained from further renement to

the allo cation algorithm for the pro jected surface area

One of the main aims of this study is to develop measures for the eectiveness of three

dimensional tree drawings In particular the conventional measure of the area of a grid drawing

has an equivalent in terms of volume when working in three dimensions However this measure

loses its relevance when the graph is drawn in a plane the volume drops to zero In addition

when considering the display of the graph on a screen it b ecomes obvious that the crucial

measure of the graphs image is the maximum size of its sides With such a measure we can

determine whether the full graph will remain visible after the application of any rotation Thus

we dene the size of a grid drawing of a graph to b e the maximum length of a side of its smallest

enclosing isothetic rectangular prism We b elieve that this measure is esp ecially relevant to

orthogonal drawings that is drawings in which edges are parallel to one of the co ordinate axes

The size of an Htree presentation for any binary tree within dimensions with n no des is O n

Theorem The size of an HTree drawing in dimensions of a complete binary tree with n

nodes is O n

The radial algorithm ab ove do es not give grid drawings and so the size measure loses its

signicance

The diameter of a graph drawing in three dimensions is the maximum distance b etween

any two vertices We prop ose two further measures for the quality of a graph drawing in three

dimensions

The ratio l where l is the length of the shortest edge and is the length of the longest

edge Long edges are dicult to follow and we b elieve a large value for this ratio charac

terises a p o or layout

The ratio d where d is the diameter of a drawing and is the shortest distance b etween

a adjacent or nonadjacent pair of no des In this case a large value is an indication of a

go o d layout

Theorem If l is the length of the shortest edge and is the length of the longest edge in a

drawing of a tree with n nodes obtained from the radial algorithm then l is O n

Theorem If d is the diameter and is the shortest distance between a pair of nodes in a

drawing of a tree with n nodes obtained from the radial algorithm then d is

Both our algorithms have b een implemented in Windows and Unix environments Samples

of binary trees have b een pro cessed against the HTree algorithm and a varied collection of trees

were pro cessed with the radial algorithm These samples give some supp ort for the prop osed

measures further we b elieve that they form an interesting collection which may b ecome useful

for b enchmarking future three dimensional tree drawing algorithms

References

F J Brandenburg Nice drawing of graphs and trees are computationally hard Technical

Rep ort MIP Fakultat fur Mathematik und Informatik University of Passau

P Crescenzi G Di Battista and A Pip erno A Note on Optimal Area Algorithms for

Upward Drawings of Binary Trees Computational Geometry Theory and Applications

P Eades Drawing free trees Technical Rep ort I IASRRE International Institute

for Advanced Study of So cial Information Science

P Eades X Lin and T Lin Two tree drawing conventions to app ear in Computational

Geometry and Applications

A Garg MT Go o drich and R Tamassia AreaEcient Upward Tree Drawings Pro c

ACM Symp on Computational Geometry

J Manning and MJ Atallah Fast detection and display of symmetry in trees Congressus

Numerantium

E Reingold and J Tilford Tidier drawings of trees IEEE Transactions on Software

Engineering SE

G GRob ertson J D Mackinley and S KCard Cone Trees Animated D visualizations

of hierarchical information Proc CHI pp

J Snyder Map Pro jections A Working Manual US Geological Survey Professional Paper

US Government Printing Oce Washington

K Sup owit and E Reingold The complexity of drawing trees nicely Acta Informatica

J S Tilford Tree drawing algorithms Masters thesis Department of Computer Science

University of Illinois at Urbana Champaign

L Valiant Universality considerations in VLSI circuits IEEE Transactions on Computers

J Vaucher Pretty printing of trees Software Practice and Experience

C Wetherall and A Shannon Tidy drawings of trees IEEE Transactions on Software

Engineering SE

Area Requirement of Visibility Representations of Trees

y z x

Go os Kant Giusepp e Liotta Rob erto Tamassia and Ioannis G Tollis

The problem of drawing a graph in the plane has received increasing attention recently due

to the large numb er of applications Examples include VLSI layout algorithm animation

visual languages and CASE to ols Vertices are usually represented by p oints and edges by simple

op en curves Another interesting representation is to map vertices into horizontal segments and

edges into vertical segments Such a representation is called a visibility representation In

this pap er we study the area requirement of various typ es of visibili ty representations of trees

and we present linear time algorithms for drawing such representations with optimal area

The concept of visibility b etween ob jects plays an imp ortant role in various problems of

computational geometry where we say that two ob jects of a given set are visible if they can b e

joined by a segment which do es not intersect any other ob ject Two ob jects of the set are visible

if they can b e joined by a nonzero thickness band which do esnt intersect any other ob ject The

ob jects are non overlapping A visibili ty representation of a graph maps vertices into ob jects

and edges into segments b etween visible vertexob jects Various visibility representations have

b een considered in the literature and received increasing attention recently see for an up to

date overview

Tamassia Tollis studied three typ es of visibility representations weak and strong

of graphs A weak visibility representation maps vertices to horizontal segments and edges to ver

tical segments having only p oints in common with the pair of horizontal segments corresp onding

to the vertices they connect Algorithms for constructing weak visibili ty representations were

presented in and indep endently in This typ e of representation has b ecome a core item

in the eld of graph drawing Recently Kant showed that such a visibility representation can

b e constructed in linear time on a grid of size at most b nc n A strong visibility rep

resentation maps vertices to horizont al segments such that two segments are visible if and only

if the corresp onding vertices are adjacent Tamassia Tollis showed that every triangular

planar graph and connected planar graph has a strong visibility representation However

deciding whether a general planar graph has a strong visibility representation is NPcomplete

Several years after the publication of the rst pap ers researchers started the study of the

dimensional variant of this problem vertices are represented by isothetic rectangles and edges

are represented by horizontal or vertical segments having only p oints in common with the pair

of rectangles corresp onding to the vertices they connect We only consider rectangles with

nonzero area and sides parallel to the xaxis and y axis This representation is called weak

visibility representation In Wismath proves that every planar graph admits a visibility

representation that is a weak visibili ty representation representation with the ad ditional

prop erty that two rectangles are visible if and only if the corresp onding vertices in the graph

Research supp orted in part by the National Science Foundation under grant CCR by the US Army

Research Oce under grant DAALG by the Oce of Naval Research and the Advanced Research

Pro jects Agency under contract NJ ARPA order and by ESPRIT Basic Research Action

No Pro ject ALCOM I I This work was p erformed in part at the Bellairs research Institute of McGill

University

Department of Computer Science UtrechtUniversity PO b ox TB Utrecht NL go os csruunl

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza Roma Italy This work

has b een done while this author was visiting the Scho ol of Computer Science of McGill University Montreal

liottainfokiting uni romai t

Department of Computer Science Brown University Providence RI rtcsbrownedu

Department of Computer Science University of Texas at Dallas Richardson TX

tollisutdall asedu

are adjacent A strong visibility representation maps each vertex to a rectangle such that two

rectangles are visible if and only if the corresp onding vertices in the graph are adjacent

Recently the area of the representation has gained a lot of attention esp ecially in the eld

of graph drawing Eades et al in study dierent p ossible representations

of trees under the constraint of minimizing the size of the drawing For a complete survey on

graph drawing see The area of a drawing is the area of the smallest rectangle with sides

parallel to the axes covering the drawing The width and the height of the drawing are the width

and the height of the covering rectangle We assume the existence of a resolution rule which

implies that the width and the height of a drawing cannot b e arbitrarily scaled down A typical

resolution rule for and visibili ty representations is requiring for the endp oints of the vertex

segments or vertex rectangles to b e placed at the p oints of an integer grid The existence of such

a resolution rule naturally raises the problem of computing and visibility representations of

a graph with minimum area

We investigate the strong visibility problem for trees The contribution is twofold First we

show lower b ounds on the area o ccupied by any and strong visibili ty representation of trees

Next we present linear time drawing algorithms that obtain such representations achieving these

b ounds Since in this pap er we only study and strong visibility representations we call them

and visibility representations for brevity

In the rest of this abstract we give a list of the main results

Theorem Let T be a rooted tree with n vertices l leaves and height h The area required by

a visibility representation T of T is h l Also T can be computed in O n time

Let T b e a free ie unro oted tree Let v b e a vertex of T and let T T b e the subtrees

v v

obtained by removing v and the edges incident on v We ro ot each subtree T at the unique

i k

vertex of T adjacent to v in T We assume that always hT hT hT in this

v v v v

pap er T is called the k th highest subtree of v

We denote with T the tree obtained by deleting from T the rst and the second subtree

of v and the incident edge of v to T and to T Ro ot T at v We call the third vertex of T

v v

the vertex v such that the height of the third highest subtree of v is maximum ie for which

hT is maximum Let k b e the corresp onding height ie k hT

v v

Theorem Let T be a free tree with n vertices and l leaves let k be the height of the third

subtree of the third vertex of T The area required by a visibility representation T of T is

k l n Also T can be computed in O n time

Theorem Let T be a free tree with n vertices l leaves and height h The area required by a

visibility representation T of T is l n Also T can be computed in O n time

Acknowledgments

This research is a consequence of the authors participation to the Workshop on Visibility Repre

sentations of Graphs organized by Sue Whitesides and Joan Hutchinson at the Bellairs Research

Institute of McGill University Feb We thank the other participants of the Work

shop for useful discussions

References

T Andreae Some Results on Visibility Graphs preprint

P Crescenzi G Di Battista and A Pip erno A Note on Optimal Area Algorithms for

Upward Drawings of Binary Trees to app ear in Comp Geometry Theory and Applications

G Di Battista P Eades R Tamassia and IG Tollis Algorithms for Automatic Graph

Drawing An Annotated Bibliography Dept of Comp Science Brown Univ Technica l

Rep ort Available via anonymous ftp from wilmacsbrownedu les

pubgdbibliotexZ and pubgdbibliopsZ

G Di Battista R Tamassia and IG Tollis Constrained Visibility Representations of

Graphs Inform Process Letters pp

G Di Battista R Tamassia and IG Tollis Area Requirement and Symmetry Display of

Planar Upward Drawings Discr and Comp Geometry pp

P Eades T Lin and X Lin Two tree Drawing Conventions Key Centre for Software

Technolog y Dept of Comp Science The Univ of Queensland Techinal Rep ort No

to app ear in Computatio nal Geometry and Applications

P Eades T Lin and X Lin Minimum Size hv Drawings Advanced Visual interfaces Pro c

eedings of AVI World Scientic Series in computer science Volume pp

G Kant A More Compact Visibility Representation in J van Leeuwen Ed Proc th

Intern Workshop on GraphTheoretic Concepts in Comp Science WG Lecture Notes

in Comp Science SpringerVerlag to app ear

J ORourke Computational geometry column Int Journal of Comp Geometry Appl

P Rosenstiehl and R E Tarjan Rectilinear Planar Layouts andBip olar Orientations of

Planar Graphs Discr and Comp Geometry pp

R Tamassia and I G Tollis A Unied Approach to Visibili ty Representations of Planar

Graphs Discr and Comp Geometry pp

SK Wismath BarRepresentable Visibility Graphs and Related Flow Problems Dept of

Comp Science Univ of British Columbia Technical Rep ort

Ecient Computation of Planar StraightLine Upward

Drawings

y y

Ashim Garg and Rob erto Tamassia

An upward drawing of a digraph is such that the edges are curves monotonically increasing in

the y direction Clearly a digraph admits an upward drawing if and only if it is acyclic Upward

drawings eectively visualize hierarchical relationships such as partial orders PERTdiagrams

and isa diagrams

In this pap er we investigate planar straightline upward drawings of digraphs We shall

denote with n the numb er of vertices of the digraph currently b eing considered Previous results

are

A digraph admits a planar straightline upward drawing if and only if it is a subgraph of

a planar stdigraph ie a planar acyclic digraph with one source and one sink joined by

an edge

A planar straightline upward drawing of a planar stdigraph can b e constructed in O n log n

time This algorithm assigns real co ordinates to the vertices and no b ound on the area of

the drawing is provided

There exists a family of planar acyclic digraphs that require exp onential area in any planar

straightline upward drawing with vertices placed at grid p oints Namely for any p ositive

integer n there exists an nvertex planar acyclic digraph G such that any planar straight

line upward drawing of G with integer vertex co ordinates has area

Our new results are summarized as follows

We give an optimal O ntime algorithm for constructing a planar straightline upward

drawing of a planar stdigraph

We present the rst NC parallel algorithm for constructing planar straightline upward

drawings Our algorithm runs in time O log n on a CRCW PRAM with n pro cessors

We show that the exp onential area lower b ound for planar stdigraphs is tight and can b e

eciently attained Namely we argue that the drawings pro duced by the aforementioned

sequential and parallel algorithms have area O c for some constant c

Both the parallel and sequential algorithms use integer arithmetic where the size of the

op erands is O A where A is the area of the drawing pro duced by the algorithm

We give a partial characterization of the area requirement of planar straightline upward

drawings of maximal planar stdigraphs based on the nesting of separating triangles

Our algorithms are based on a technique that rst contracts a large nsize subset of

edges then recursively computes a drawing of the resulting subgraph and nally restores the

contracted edges to yield a drawing of the original digraph

Research supp orted in part by the National Science Foundation under grant CCR by the US Army

Research Oce under grant DAALG and by the Oce of Naval Research and the Defense Advanced

Research Pro jects Agency under contract NJ ARPA order

Department of Computer Science Brown University Providence RI fagrtgcsbrownedu

References

G Di Battista and R Tamassia Algorithms for Planar Representation of Acyclic Digraphs

Theoretical Computer Science vol pp

G Di Battista R Tamassia and IG Tollis Area Requirement and Symmetry Display of

Planar Upward Drawings Discrete Computational Geometry vol no pp

An Approach for BendMinimal Upward Drawing

Uli Fomeier and Michael Kaufmann

Besides minimizing the area of the emb edding and edge lengths the minimization of the

numb er of b ends is the measure of the quality of a planar emb edding The only provably optimal

algorithm for this criterion of Tamassia reduces the problem to a mincostow problem and

has therefore a quite high runtime of O n log n Several heuristics are must faster

but do not guarantee optimality

Our approach has the goal to achieve optimality in a simple way for a certain class of

emb eddings namely upwarddrawings We show how to get optimal b ounds on the numb er of

b ends in linear time Upward drawings are a p opular way to display acyclic digraphs such that

all edges ow in the same direction from b ottom to top Most algorithms for upward drawings

for planar graphs concern the minimization of the area display of symmetries and isomorphisms

The problem of rectilinear upward drawing do es not arise in the literature so often a variation

app ears in

We dene a rectilinear drawing to b e upward if all segments of the edges are either horizontally

or upward and for each no de v with incomingoutgoing edges one incomingoutgoing edge is

incident to v by a vertical segment As in we rst determine a top ological emb edding of the

graph namely the direction each edge starts in and the sequence of b ends on it After this a

linear standard compaction algorithm computes the nal emb edding of the graph in the grid

Here we describ e only the rst step We assume that the graph is a subgraph of a planar

stgraph Rectilinear upward drawings exists only for those graphs

Basic Ideas

Before describing the algorithm we start with some observations Let G V E b e an

acyclic planar st graph with maximum degree emb edded in the plane Every edge has two

directions a start direction the way how the edge leaves a no de and an end direction the

way how the edge joins a no de Valid directions are only left upward and right If the start

direction and the nish direction of an edge are dierent the edge gets one or two b ends

Our algorithm assigns appropriate directions for the start and end segments to all edges and

puts the fragments together by pro ducing b ends if necessary

At any no de there are at most two dierent ways of assigning the edges ie the connections

to the neighb ours to the pins left upward right and downward If a no de has eg two exits

we can assign them either to the upp er and right pin or to the left and upp er pin If we make

the wrong decision there may b e diculties to connect the remaining edges in the example

WilhelmSchi ckardInsti tut fur Informatik Universitat Tubingen Sand Tubingen Germany

mkinformatikunituebi ngen de

the entries Therefore it is imp ortant to determine an order how to do the assignment Clearly

there is no problem with no des with one or three entries or exits resp ectively In the former

case it has to b e the lower upp er pin in the latter case the left lower and right right upp er

and left pin in this order No des of degree or with two entries and two exits are called critical

For critical no des we always have two choices

It is well known that the incoming edges of each no de v of a planar stgraph app ear

consecutively around v and so do the outgoing This makes sense to say that an exit entry is

at the left or at the right of another

Lemma Let v b e a no de of degree with two exit edges e and e st e is left of e An

i j i j

arbitrary assignment of e to a pin increases the numb er of b ends on e by at most one There

i j

are no global eects on the emb edding ie the neighb ours of v do not realize the dierence

An analogous fact holds for critical no des with two entries

Pro of Let c b e the numb er of b ends in the case that the left and upp er pins are the exit

pins of v We can simulate the other conguration upp erright pins are exits by turning the

edge out of the upp er pin to the left and the edge out of the right pin upward using two new

b ends

Lemma Let v b e a critical no de of degree For every pair e e of adjacent edges their

i j

b endcosts increase by at most two if e and e are assigned arbitrarily to pins The situation of

i j

the neighb ours is not changed thereby no global eects

As already seen computing start and end directions of an edge connecting two noncritical

no des is easy For an edge b etween a critical no de v of degree and a noncritical no de Lemma

ensures that it is always go o d to cho ose the cheap er way to connect the noncritical no de

b ecause the other entry exit pin of v requires at most one b end more than its neighb our pin to

connect any edge Thus the chosen solution is not worse than the alternative After this action

v b ecomes noncritical its other entry exit pin has no more choice

The case of an edge from a noncritical no de u to a critical no de v of degree is more

complicated since three pins are inuenced by xing one thus it is not go o d to apply the greedy

strategy here

Denition A critical no de v is called maiden if less than two of its neighb ours have already

xed their connection to v If the two alternatives of the connections to the xed neighb ours

have dierent costs v is called decided and tie otherwise

Note that a tie no de always has degree Lemma tells us that if a no de is decided the

dierence b etween the two alternatives is exactly two b ends With the argument ab ove we

can cho ose the cheap er one in such cases b ecause the two other pins require at most two b ends

more than in their optimal layout

If there are no more decided no des ie all no des still unxed are tie we compute a component

M V of no des such that every v M is tie M is connected in the underlying undirected

graph and jM j is maximal We take a no de on the b order of M ie connected with two no des

M and assign its pins in an arbitrary way Thereby its neighb ours in M b ecome decided

and so on Step by step the whole comp onent can b e xed Note that the state of the critical

no des may change after each step

The algorithm follows the description of the dierent cases ab ove The no des are partitioned

into dierent classes critical noncritical and the edge directions are assigned by lo cal

insp ections The missing comp onents at the end are handled as describ ed ab ove

The correctness of the algorithm follows from the fact that it only determines a connection

if either the alternative cannot b e b etter noncritical no des decided no des or the alternatives

have b een proved to b e equal comp onents of tie no des

The Algorithm

At each step of the algorithm we have to determine the new state of at most four critical no des

This can b e done in constant time The rst two phases noncritical no des and decided no des

are trivially linear The only diculty is to compute the comp onents M in phase But this

is easy to o starting with an actual subset M of V at the b eginning one tie no de we have to

test the neighb ours of no des in M if they are tie Therefore we make at most jM j tests All

comp onents are disjoint follows from the denition of a comp onent therefore we have to make

O n tests

Theorem The algorithm works in linear time

Remarks

The class of rectilinear upward drawings considered here is a prop er sub class of the class

commonly known as rectilinear upward drawings each edge is represented by a monotonically

increasing curve If we consider the dierences b etween the two mo dels we nd examples with

much more b ends for our mo del But it seems that these unnecessary b ends can b e removed very

easily With small mo dications our algorithm is a go o d heuristic for the problem of general

rectilinear upward drawing

References

Di Battista G R Tamassia and IG Tollis Area requirement and symmetry display in

drawing graphs Discrete and Comp Geometry pp

Storer JA On minimal no decost planar emb eddings Networks pp

Tamassia R On emb edding a graph in the grid with the minimum numb er of b ends SIAM

J Comput pp

Tamassia R and IG Tollis A Unied Approach to Visibility Representations of Planar

Graphs Discrete and Comp Geometry pp

Tamassia R and IG Tollis Ecient emb edding of planar graphs in linear time in Proc

IEEE Int Symp on Circuits and Systems Philadelphia pp

Representations of Planar Graphs

C Thomassen

Abstract not Available

On Lattice Structures Induced by Orientations

Patrice Ossona de Mendez

First we remark that an acyclic orientation of a ro oted graph denes a distributive lattice

on its algebraic co circuits ie oriented cuts Let v b e the ro ot of an acyclic oriented graph G

recall that an algebraic co circuit of G may b e expressed as a sum of vertex co circuits with integer

co ecients If we constrain the v co cyles co ecient to b e null the decomp osition is unique

and denes an injection from the co circuits into a ZZ free mo dule The total order of ZZ denes

a partial order on the algebraic co circuits of G This partial order is a distributive lattice the

inmum and supremum op erators are expressed in terms of min and max on the co ecients

The Hasse diagram of a lattice is the oriented graph whose vertices are the elements of the lattice

and whose edges corresp ond to the immediatesucessor relation In the Hasse diagram of the

co circuits lattice two co circuits are adjacent if and only if their distance is in the ZZ mo dule

By duality the algebraic circuits of an oriented facero oted planar graph have also a distributive

lattice structure This circuits lattice seems to b e more p owerfull

We shall rst give an example where this circuits lattice structure dened on the ebip olar

orientations leads to relevant geometric interpretations Thatfor we express a bijection b etween

ebip olar orientations and algebraic circuits of a planar graph

Given a biconnected graph G and an oriented edge e s t of G an ebip olar orientation

of G is an acyclic orientation of the edges of G having s as a unique source and t as a unique

sink

Bip olar orientations are closely related to connexity and planarity They have

b een extensively used to p erform graph drawing of planar graphs upward

drawings and testing graph planarity

In the planar case the angle graphe AG of G also called radial graph is dened as the

vertexface adjacency graph related to an emb edding of G The ebip olar orientations of G

are in bijection with the edge colorations of AG satisfying lo cal conditions Given one such

coloration the ebip olar orientations are in bijection with the alternating cycles

An edge coloration of a bipartite graph naturally denes an orientation of the edges map

ping alternating cycles into oriented circuits The ebip olar orientations of G are thus in bijection

with the circuits of an induced orientation of AG If the graph is connected each edge of

AG except those incident to the vertices corresp onding to vertices and faces of G incident to

e b elongs to a circuit Hence the graph AG is totaly cyclic and by duality we may apply

the lattice construction dened previously the set of the ebip olar orientations of a connected

plane graph G is a distributive lattice and its Hasse diagram is the adjacency graph of the e

bip olar orientations of G two orientations b eing adjacent if they dier by the orientation of a

unique edge The connexity of the lattice implies the connexity of the adjacency graph of the

ebip olar orientations of a connected This last result holds also for non planar connected

graphs as proved in The minimum and maximum ebip olar orientations of the lattice

have a very simple geometric interpretation exhibited by the left and the right pathpacking

algorithms

The ebip olar orientations of a connected graph G are in bijection with its stupward

drawings The Hasse diagram describ es allowable lo cal deformations of an upward drawing

The connexity implies that any two upward drawings can b e derived from one another through

successive lo cal deformations

Ecole des Hautes Etudes en Sciences So ciales Boulevard Raspail PARIS

As an other application we mention the lattice structure of the trees decomp ositions

intro duced by W Schnyder and C Thomassen In this later case the successor relation is related

to a circular p ermutation of the tree assignments among a triangle This lo cal transformation

generates the lattice and hence all the trees decomp ositions

References

M Bousset Orientation dun schema par passage dun ot dans les angles PhD thesis

Ecole des Hautes Etudes en Sciences So ciales Paris

H de Fraysseix and P O de Mendez On tree decomp ositions and angle marking of planar

graphs in preparation

H de Fraysseix and P O de Mendez Planarity and edge p oset dimension submitted to

the Europ ean Journal of Combinatorics

H de Fraysseix P O de Mendez and J Pach Representation of planar graphs by segments

to app ear in Intuitive Geometry

H de Fraysseix P O de Mendez and P Rosenstiehl Bip olar orientations revisited In

Fifth FrancoJapanese Days on Combinatorics and Optimization

H de Fraysseix P O de Mendez and P Rosenstiehl On triangle contact graphs In

Cambridge combinatorial Conference in honor of Paul Erdos

G Di Battista and R Tamassia Algorithms for plane representations of acyclic digraphs

Theoret Comput Science

G Kant Hexagonal grid drawings Internal rep ort of the Utrecht University

A Lemp el S Even and I Cederbaum An algorithm for planarity testing of graphs In

Gordon and Breach editors Theory of Graphs pages

P O de Mendez Orientations bipolaires PhD thesis Ecole des Hautes Etudes en Sciences

So ciales Paris To b e defended

P O de Mendez The plane bip olar orientations lattice In Sixth FrancoJapanese Days

on Combinatorics and Optimization

P Rosenstiehl Emb edding in the plane with orientation constraints the angle graph

Ann NY Acad Sci pages

P Rosentiehl and RE Tarjan Rectilinear planar layout and bip olar orientation of planar

graphs Discrete and Computational Geometry

R Tamassia and IG Tollis A unied approach to visibility representations of planar

graphs Discrete Comput Geom

Complexity of Intersection Classes of Graphs

Jan Krato chvl and Jir Matousek

Many of the well known and studied classes of intersection graphs of geometrical ob jects in

the plane p ossess nice prop erties with resp ect to algorithmic complexity Often such problems

like stable set clique or chromatic numb er are p olynomially solvable and many of such classes

are recognizable in p olynomial time cf interval graphs circular arc graphs circle graphs

function graphs etc On the other hand already slightly more general classes are NPhard

to recognize eg intersection graphs of straight segments Here we give a brief survey of the

complexity of the recognition problem

If C is a class of sets geometric ob jects in our case then the class of intersection graphs of

C denoted by IGC will b e the class of all simple undirected graphs isomorphic to graphs of

the form G V E where V C and e uv E i u v We call this V a representation

of the isomorphism class of G

We set all the ob jects in consideration are planar

S T RI N G IGfall simple curvesg

CONV IGfall convex setsg

SEG IGfall straight line segmentsg

k DIRd d IGfall segments with slop es among d d g

k k

d d real numb ers

k DIR fk DIRd d d d real numb ersg

k k

Note that in our setting interval graphs IGfsegments of a lineg circular arc graphs

IGfsegments of a circleg circle graphs IGfchords of a circleg and all of these classes can

b e recognized in p olynomial time

Theorem i Recognition of STRING graphs is NPhard no upper bound is known ii

Recognition of CONV and SEG graphs is NPhard and both these problems are in PSPACE

iii For every xed k recognition of k DIR graphs is NPcomplete

iv Recognition of k DIRd d graphs is NPcomplete note that here k and d d

k k

are part of the input

v Intersection graphs of isothetic rectangles ie graphs of boxicity are NPcomplete to

recognize

vi Recognition of SEG graphs is polynomial ly equivalent to deciding solvability of a system of

strict polynomial inequalities in real numbers

Unlike most decesion problems of similar nature the recognition problems treated in The

orem are not known to b e in NP cv parts iii or their memb ership in NP is nontrivial

parts iiiiv The following results show why the usual guess and check scheme of proving

NPmemb ership fails for STRING SEG and k DIR graphs

Prague Czech Republic

Theorem i For every n there is a graph G S T RI N G on O n vertices such that in

each of its STRINGrepresentations there are two curves which share at least crossing points

ii For every n there is a graph G SEG on O n vertices such that in each of its

SEGrepresentations whose segments have integer endpoints there is segment with an endpoint

coordinate of size at least

iii For every n there is a graph G DIR on O n vertices such that in each of

its DIRrepresentations whose segments have integer endpoints there is a segment with and

endpoint coordinate at least

References

JKrato chvl J Matousek NPhardness results for intersection graphs Comment Math

Univ Carolin MR c

JKrato chvl String graphs I I Recognizing string graphs is NPhard J Combin Theory

Ser B MR g

JKrato chvl J Matousek String graphs requiring exp onential representations J Com

bin Theory Ser B MR g

JKrato chvl A sp ecial planar satisability problem and some consequences of its NP

completeness Discrete Appl Math to app ear

JKrato chvl J Matousek Intersection graphs of segments J Combin Th Ser B to

app ear

On Triangle Contact Graphs

y y y

Hub ert de Fraysseix Patrice Ossona de Mendez and Pierre Rosenstiehl

Various representations of the same planar map are describ ed b elow Vertices edges and

faces are alternatively represented by p oints arcs or disks of the plane By representation of a

planar map it is understo o d that the circular order of the edges around each vertex is preserved

An old problem of geometry consists of representing a planar map M by a collection of disks

matched with the vertices of M These disks are disjoint except at contact p oints for some

pairs of them the contacts representing the edges of M The case of unconstrained disks is

merely solved by drawing at each vertex p oint v a closed curve surrounding v and half of the

edges arcs incident to v in a tubular way The diculty starts when the disks have to resp ect

a prescrib ed shap e The famous case of circular disks known as the AndrewThurston circle

packing Theorem hits diculties of numerical analysis it has b een improved to p olynomial

complexity and generalized recently

We are concerned here with triangular disks such that each contact p oint is a vertex of

one triangle and b elongs to the side of another one This asymetry induces an orientation of

the contact that is an orientation of the corresp onding edge Such an arrangement is called a

This work was partially supp orted by the ESPRIT Basic Research Action Nr ALCOM I I

CNRS EHESS Boulevard Raspail Paris France

triangle contact system So it is obvious that to any triangle contact system S is asso ciated

an oriented planar map M S We assume b elow that the orientation of the contact edge is

towards the triangle which gives a vertex in the contact

A rst result is that any planar map may b e represented by a triangle contact system The

result still holds if we imp ose a coloration to the triangles and the contacts in the following way

the edges taken in clo ckwise order are colored resp ectively a b and c and each vertex is colored

as its opp osite side now a contact has to b e b etween a vertex and a side of the same color

An instance of this orientation constraint consists of forcing the triangles to have a common

oriented angle in the ane plane

In the case of maximal planar maps M the coloration of the triangles is intrinsicaly imp osed

We intro duce a coloration that partitions the edges not incident to the innite face in three

oriented trees ro oted on the frame which are nothing else but the Schnyder trees of a maximal

planar map A trees decomp osition of Schnyder or a Schnyder orientation is dened as an

orientation of the edges b esides the innite face and a coloration of them in three colors a b

and c such that at each vertex b esides those of the innite face there is exactly three incoming

edges colored resp ectively a b and c in clo ckwise order and such that the outgoing edges of

one color are group ed in the angle formed by the two incoming edges of the other colors By

denition each color denes a tree ro oted on the innite face

The construction of a colored triangle contact system representing a planar map is achieved

in linear time and space considering that arithmetic is p erformed in constant time over the

rational eld

A variation of our main result is that any maximal planar map may b e represented by an

isosceles triangle contact system each having an horizontal basis and the opp osite vertex placed

b elow

Let a Tshap e in the O x O y plane b e a pair of an horizontal segment and a vertical one

placed b elow with a contact p oint A Tcontact system is a collection of Ts all disjoints

except for contact p oints consisting of an extreme p oint from one exactly and a side p oint

from the other We show how a colored triangle contact system is transformed into a Tcontact

system which provides very compact representations of planar graphs

A rectilinear representation of a planar map represents each vertex by an horizontal

segment and each edge by a vertical segment incident to two horizontal vertex segment

In each face of M is represented by a disk with a lowest vertex and a highest vertex segment

a O y monotone left b oundary and a O y monotone right b oundary this one b eing a straight

segment Therefore by extending each vertex segment on its right up to the rst met vertical

segment the representation of M b ecomes a partition of a rectangle into rectangles each

vertex is represented by an horizontal segment each face by a vertical segment and each edge

by a rectangular disk Such a representation is called a tessalation representation of M Any

connected planar map has a tessalation representation as also explained by Tamassia and

Tollis

Actually all the tessalation representations of a connected graph are obtained by using the

bipartite planar graph representation of the radial graph by contact segments as dened in

We get here a tessalation representation of a maximal planar map from an interesting feature

of the triangle contact representation of a maximal planar map M that is the representation

of the faces of M by triangles In case each vertex triangle is isosceles with its basis parallel to

O x and the opp osite vertex placed b elow each face is a triangle with each basis parallel to

O x and the opp osite vertex placed ab ove By drawing a vertical height segment of a prop er

length at each triangular face and extending the horizontal segments of the vertex bases we

obtain a tessalation representation of M

References

EM Andreev On convex p olyhedra in lobacevskii spaces Mat Sb

H de Fraysseix and P O de Mendez On tree decomp ositions and angle marking of planar

graphs to app ear

H de Fraysseix P O de Mendez and J Pach A streamlined depthrst search algorithm

revisited to app ear

H de Fraysseix P O de Mendez and J Pach Representation of planar graphs by segments

to app ear in Intuitive Geometry

H de Fraysseix J Pach and R Pollack Small sets supp orting Fary emb eddings of planar

graphs Combinatorica Sto c Conference

P Eades and R Tamassia Algorithms for drawing planar graphs an annotated bibliogra

phy Tech Rep No CS Brown University

B Mohar Circle packings of maps in p olynomial time to app ear

P Rosentiehl and RE Tarjan Rectilinear planar layout and bip olar orientation of planar

graphs Discrete and Computational Geometry

W Schnyder Planar graphs and p oset dimension Order

R Tamassia and IG Tollis Tessalation representation of planar graphs In Proc Twenty

Seventh Annual Al lerton Conference on Communication Control and Computing pages

Characterisation and Construction of the Rectangular Dual of

a Graph

Simone Pimont and Michel Terrenoire

Intro duction

Circuit placement consists in p ositionning rectangular blo cks dened by their area and inter

connected by nets on a rectangular surface to b e minimized An available strategy deals with

the rectangular dualization of a connectivity graph We intro duce the notion of p olarity index

then the concept of p olarization in order to characterize graphs with rectangular duals and to

build an asso ciated rectangular representation

The three steps of placement

The minimization ob jective is approached by the construction of an initial placement step

such as the sum of the lengths of the connections is minimum This initial placement is obtained

by means of a data analysis metho d a factorial analysis on a distance matrix It denes in R

a set X of p oints corresp onding to the blo cks From this placement we dene a connectivity

graph GX that resumes the proximity relations b etween the corresp onding p oints Then by

dualization a rectangular representation asso ciated to GX is researched step that is to

say a dissection of a rectangle into rectangles for which GX describ es the adjacency relations

among the rectangles In step an iterative construction of rectangular placements is pro cessed

whose ob jective is area minimization FOU The related work fo cuses on step particularly

on the characterization and on the construction of a rectangular graph

A dualization pro cess to construct a rectangular graph

We know that a necessary condition for a planar graph to admit a rectangular dual is that it is

triangular every face except the outer one is a triangle KOZ BHA So we consider the

DELAUNAY graphe DX asso ciated to the set X However the ab ove condition is not sucient

A triangular graph admits a rectangular dual if an only if it do es not contain complex triangle

KOZ Moreover from a triangular graph one can built more than one rectangular dual

an edge b etween two no des in the graph can corresp ond to a vertical or horizontal b oundary in

the rectangular dual To take into account this phenomenon we intro duce indices of p olarity

for the edges PIM The indices can take two values horizontal or vertical We dene an

heuristic that assigns a p olarity index to each edge of DX in order to construct a rectangular

dual The elimination of the complex triangles is attempted at the same time A program was

realized in PASCAL on PC and in C on APOLLO But the ab ove p olarity do es not account by

itself of the rectangularization So we intro duce another concept that deep ens this notion

Lab oratoire dIngenierie des Systemes dInformation pimontlisisu nuni vl yon fr

Lab oratoire des metho des et analyses des systemes et des structures Universite Claude Bernard LYON

Boulevard du Novembre F Villeurban ne Cedex FRANCE

Characterization of a dualizable graph Polarizable graph

We consider a triangular graph HXW It admits a completion graph HcXcWc intro

ducing the four external vertices North East South West see KOZ for the denition of

completion

Notations

Z is the set Wc without the four edges NO OS SE EN

is a mapping of Z into the set fvhg v for vertical and h for horizontal

is a mapping named orientation of Z into Xc xy is the origin of xy

Hc is the oriented graph deduced from the graph XcZ by

a path p in Hc is a vpath resp hpath if w v resp h for every edge w

corresp onding to an arc of p

N O S E are the set of the vpaths from N to x the set of the hpaths from O to

x x x x

x the set of the vpaths from x to S the set of the hpaths from x to E

We supp ose that for every x in X satises

Nx Z Nx N and Nx v

Ox Z Ox O and Ox h

Sx Z Sx x and Sx v

Ex Z Ex x and Ex h

is a p olarization for Hc if for every x in X

N O S E are not empty

x x x x

n N o O s S e E the four sets n o n s n e o s o

x x x x

e s e are reduced to x

Then we can dene

A graph H is p olarizable if it is triangular and if it admits a completion graph Hc with

an asso ciated p olarization

For each no de x we dene a lo cal p olarization for x based on lo cal prop erties ab out

the existence and the clo ckwize order of the oriented edges admitting x as endp oints

We state two theorems PIM

Theorem

A triangular graph admits a rectangular dual if an only if it is p olarizable

Theorem

For a triangular graph H with an asso ciated triangular completion Hc is a p olar

ization if and only if is a lo cal p olarization for each vertex in Hc

Conclusion

The concept of p olarity index allows us to realize a program rather ecient over our examples

but this approach is heuristic The concept of p olarizable graph in view of the theorems and

will enable the development of more ecient algorithms for circuit placement For example

we prop ose to cho ose a couple candidate for p olarization and then to mo dify it using the lo cal

prop erties asso ciated to the no de admissible p olarization Our theoretical and practical results

allow us to argue that the p olarization and the asso ciated prop erties give a p ertinent framework

to elab orate ecient dualization algorithms

References

BHA J Bhasker and S Sahni A Linear Algorithm to Find a Rectangular Dual of a Planar

Triangulated Graph Algorithmica vol pp

FOU DFourre and S Pimont Heuristique de rectangularisation de graphe p our le place

ment de circuits rapp ort de recherche Lab oratoire Informatique de lUniversite de

Lyon

KOZ K Kozminshi and E Kinnen Rectangular duals of planar graph Networks

vol pp

PIM S Pimont and M Terrenoire Dual rectangulaire dun graphe p our le placement de

circuit TSI vol pp

Two Algorithms for Finding Rectangular Duals of Planar

Graphs

Go os Kant Xin He

The problem of drawing a graph on the plane has received increasing attention due to a

large numb er of applications Examples include VLSI layout algorithm animation visual

languages and CASE to ols Vertices are usually represented by p oints and edges by curves In

the design of o or planning of electronic chips and in architectural design it is also common to

represent a graph G by a rectangular dual dened as follows A rectangular subdivision system

of a rectangle R is a partition of R into a set fR R R g of nonoverlapping rectangles

such that no four rectangles in meet at the same p oint A rectangular dual of a planar graph

G V E is a rectangular sub division system and a onetoone corresp ondence f V

such that two vertices u and v are adjacent in G if and only if their corresp onding rectangles

f u and f v share a common b oundary In the application of this representation the vertices

of G represent circuit mo dules and the edges represent mo dule adjacencies A rectangular dual

provides a placement of the circuit mo dules that preserves the required adjacencies

This problem was studied in KK Bhasker and Sahni gave a linear time algorithm to

construct rectangular duals The algorithm is fairly complicated and requires many intriguing

pro cedures The co ordinates of the rectangular dual constructed by it are real numb ers and b ear

no meaningful relationship with the structure of the graph This algorithm consists of two ma jor

steps constructing a socalled regular edge labeling REL of G and constructing the

rectangular dual using this lab eling A simplication of step is given in The co ordinates of

the rectangular dual constructed by the algorithm in are integers and carry clear combinatorial

meaning However the step still relies on the complicated algorithm in A parallel

implementation of this algorithm working in O log n log n time with O n pro cessors is given

Department of Computer Science Utrecht University Padualaan CH Utrecht the Netherlands

go oscsruunl Research supp orted by the ESPRIT Basic Research Actions program of the EC under contract

No pro ject ALCOM I I

Department of Computer Science State University of New York at Bualo Bualo NY United States

of America xinhecsbualoedu Research supp orted by National Science Foundation grant numb er CCR

In this pap er we present two linear time algorithms for nding a regular edge lab eling The

two algorithms use totally dierent approaches and b oth are of indep endent interests The rst

algorithm is based on the edge contraction technique which was also used for drawing triangular

planar graphs on a grid The second algorithm is based on the canonical ordering for

connected planar triangular graphs This technique extends the canonical ordering which was

dened for triangular planar graphs and triconnected planar graphs to this class of graphs

Another interesting representation of planar graphs is the visibility representation which maps

vertices into horizotnal segments and edges into vertical segments It turns out that the

canonical ordering also gives a reduction of a factor in the width of the visbility representation

of connected planar graphs

References

Bhasker J and S Sahni A linear algorithm to check for the existence of a rectangular

dual of a planar triangulated graph Networks pp

Bhasker J and S Sahni A linear algorithm to nd a rectangular dual of a planar trian

gulated graph Algorithmica pp

Di Battista G P Eades R Tamassia and IG Tollis Algorithms for Automatic Graph

Drawing An Annotated Bibliography Dept of Comp Science Brown Univ Technical

Rep ort

Fraysseix H de J Pach and R Pollack How to draw a planar graph on a grid Combina

torica pp

He X On nding the rectangular duals of planar triangulated graphs SIAM J Comput

to app ear

He X Ecient Paral lel Algorithms for two Graph Layout Problems Technical Rep ort

Dept of Comp Science State Univ of New York at Bualo

Kant G Drawing planar graphs using the l mcordering Proc th Ann IEEE Symp on

Found of Comp Science Pittsburgh pp

Kozminski K and E Kinnen Rectangular dual of planar graphs Network pp

Rosenstiehl P and R E Tarjan Rectilinear planar layouts and bip olar orientations of

planar graphs Discr and Comp Geometry pp

Schnyder W Emb edding planar graphs on the grid in Proc st Annual ACMSIAM

Symp on Discr Alg San Fransisco pp

Tamassia R and I G Tollis A unied approach to visibility representations of planar

graphs Discr and Comp Geometry pp

A More Compact Visibility Representation

Go os Kant

Intro duction

The problem of nicely drawing a graph in the plane has received increasing attention due

to the large numb er of applications Examples include VLSI layout algorithm animation

visual languages and CASE to ols Several criteria to obtain a high aesthetic quality have b een

established Typically vertices are represented by distinct p oints in a line or plane and are

sometimes restricted to b e grid p oints Alternatively vertices are sometimes represented by

line segments Edges are often constrained to b e drawn as straight lines or as a contiguous set

of line segments eg when b ends are allowed The ob jective is to nd a layout for a graph

that optimizes some cost function such as area minimum angle numb er of b ends or satises

some other constraint see for an up to date overview

One of the most b eautiful ways for drawing G is by using a visibility representation In

a visibility representation every vertex is mapp ed to a horizontal segment and every edge

is mapp ed to a vertical line only touching the two vertex segments of its endp oints It is

clear that this leads to a nice and readable picture and it therefore gains a lot of interest

It has b een applied in several industrial applications for representing electrical diagrams and

schemas Rosenstiehl p ersonal communication Otten Van Wijk showed that every

planar graph admits such a representation and a linear time algorithm for constructing it is

given by Rosenstiehl Tarjan indep endently Tamassia Tollis came up with the same

algorithm The size of the required grid is n n with n the numb er of vertices

The algorithm is based on a so called stnumbering a numb ering v v of the vertices such

that v v G and every vertex v i n has neighb ors v and v with j i k The

n i j k

height of the drawing is the longest path from v to v which has length at most n The

width of the drawing is the longest path in the dual graph which is f where f is the numb er

of faces in G by Eulers formula m n and f m n

The algorithm is used in several drawing algorithms We mention here the algorithm of

Tamassia Tollis for constructing an orthogonal drawing and the work of Di Battista

Tamassia Tollis for computing constrained visibility representations Rosenstiehl Tarjan

also discuss the op en problems concerning the grid size of visibility representations The

requirement of using a small area seems to b ecome a core area in the research eld of graph

drawing due to the imp ortant applications in VLSIdesign and chip layout eg see Kant

In this pap er we show that every planar graph can b e represented by a visibility representa

tion on a grid of size at most b nc n This improves all previous b ounds considerably

An outline of the algorithm to achieve this is as follows Assume the input graph G is triangu

lated otherwise a simple linear time algorithm can b e applied to make it so Then we split

G into its connected comp onents and construct the blo ck tree of G We show that we can

do this in linear time for triangulated planar graphs thereby improving the O n m n m

time algorithm of Kanevsky et al for this sp ecial case To each connected comp onent the

algorithm of Kant He is applied who showed that if the planar graph is connected then a

This work was supp orted by ESPRIT Basic Research Action No pro ject ALCOM I I Algorithms and

Complexity Part of this work was done while visiting the Graph Theory workshop at the Bellairs Research

Institute of McGill University Montreal Feb

Dept of Computer Science Utrecht University Padualaan CH Utrecht the Netherlands

go oscsruunl

visibility representation of it can b e constructed with grid size at most n n The

representations of the connected comp onents are combined into one entire drawing leading to

the desired width

References

Chiba N and T Nishizeki Arb oricity and subgraph listing algorithms SIAM J Comput

Di Battista G Eades P and R Tamassia Algorithms for Automatic Graph Drawing An Anno

tated Bibliography Dept of Comp Science Brown Univ Technical Rep ort

Di Battista G R Tamassia and IG Tollis Constrained visibility representations of graphs Inform

Process Letters pp

Kanevsky A R Tamassia G Di Battista and J Chen Online maintenance of the fourconnected

comp onents of a graph in Proc th Annual IEEE Symp on Found of Comp Science Puerto

Rico pp

Kant G Drawing planar graphs using the l mcordering Proc th Ann IEEE Symp on Found

of Comp Science Pittsburgh pp

Revised and extended version in

Kant G Algorithms for Drawing Planar Graphs PhD thesis Dept of Computer Science Utrecht

University

Kant G On triangulating planar graphs submitted to Information and Computation

Kant G and X He Two Algorithms for Finding Rectangular Duals of Planar Graphs Tech Rep ort

RUUCS Dept of Computer Science Utrecht University

Otten RHJM and JG van Wijk Graph representation in interactive layout design in Proc

IEEE Int Symp on Circuits and Systems pp

Rosenstiehl P and RE Tarjan Rectilinear planar layouts and bip olar orientations of planar graphs

Discr and Comp Geometry pp

Tamassia R and IG Tollis A unied approach to visibility representations of planar graphs Discr

and Comp Geometry pp

Tamassia R and IG Tollis Planar grid emb edding in linear time IEEE Trans Circuits and

Systems pp

Cone Visibility Graphs

Anna Lubiw

Intro duction

Two main ways of representing graphs are as intersection graphs and as visibili ty graphs using the

symmetric relations of intersection and visibility resp ectively See G for a survey on intersection

graphs and OR for a survey on visibili ty graphs A main dierence b etween the two is that

visibility usually carries with it the notion of blo cking consequently the classes of graphs are

usually not closed under induced subgraphs and are more dicult to characterize

This work explores a class of directed graphs dened by means of an asymmetric relationship

which has some of the features of intersection and some of the features of visibility Let P b e

a set of p oints in the plane With each p oint p let there b e an ordered pair of rays r p r p

emanating from the p oint determining the closed clo ckwise cone C p going from r p to r p

The cone visibility digraph of this conguration is a directed graph on vertex set P with an edge

from vertex p to vertex p i p is contained in the cone C p A digraph is a cone visibility

digraph if it has such a representation for some choice of p oints and cones

These are visibili ty graphs in the sense that the cone at a p oint determines which other

p oints it can see There are two dierences b etween this notion and previous denitions of

visibility graphs one is that the relationship is asymmetric so the graphs are directed another is

that there is no notion of blo ckingwhether one p oint sees a second p oint is indep endent of the

other p oints There are two signicant consequences of this One is that cone visibility graphs are

closed under induced subgraphsthus there might b e b e a forbidden subgraph characterization

of the class Another consequence is that given a representation of a cone visibili ty graph there

is an O test for whether two p oints are joined by an edgethus a cone visibili ty representation

of a graph is a very ecient representation

Relationship to visibility graphs of p olygons

Un directed cone visibility graphs can b e obtained by symmetrizing the relationship put an

undirected edge b etween p and p i p is in the cone C p and p is in the cone C p These

graphs have enough relation to visibility graphs of p olygons that an understanding of them

may help us understand visibility graphs of p olygons Given a simple p olygon in the plane the

vertices and edges determine a cone system in the obvious way The undirected cone visibility

graph contains all the edges of the visibility graph of the p olygon plus edges joining vertices

whose visibility in the p olygon is blo cked by an edge not incident with either vertex

Relationship to dimension of p osets

Whenever the rays r p are all parallel for all p oints p and the rays r p are all parallel

and the angle b etween them is no more than then the resulting cone visibility digraph is

a p oset See T for information on p osets In particular when all the rays r p are parallel

to the p ositive y axis and all the rays r p are parallel to the p ositive x axis the realizable

cone visibili ty digraphs are exactly the p osets of dimension A p oset has dimension n if it

Research supp orted by NSERC Department of Computer Science University of Waterlo o Waterlo o Ontario

Canada NL G alubiwuwaterlo oca

can b e realized in ndimensional space as the visibility digraph of cones that are translates of

the p ositive quadrant but cannot b e so realized in a lowerdimensional space The bipartite

graph S with vertices v v and u u and an edge from v to u i i j is a standard

n n n i j

example of a p oset of dimension n Though S for n do es not have dimension it is a

cone visibility digraph in the plane

Positive results

Among the undirected graphs that are cone visibili ty graphs are trees cycles complete graphs

complete bipartite graphs and bipartite p ermutation graphs These results are not dicult and

are proved by giving constructions Turning to digraphs it is not dicult to show that trees

and cycles with arbitrary directions on the edges are cone visibility digraphs

Negative results

This section is joint work with Jonathan F Buss One example of a graph that is not a cone

visibility digraph is the bipartite graph with vertex set V V where jV j and V has a

vertex corresp onding to each of the triples of V with edges to those three vertices of V To

prove that this graph is not a cone visibility digraph it suces to prove that it is not p ossible to

arrange p oints in the plane so that every triple can b e separated from the remaining p oints by

a cone We cannot have p oints forming a convex gon otherwise the three o dd p oints around

the gon cannot b e separated We also cannot have three or more convex hulls in the onion

p eeling of the p oint set otherwise three p oints of the middle convex hull cannot b e separated

But with or more p oints we cannot avoid b oth these situations

Op en questions

Characterize cone visibility digraphs andor nd a go o d recognition algorithm

References

G MC Golumbic Algorithmic Graph Theory and Perfect Graphs Academic Press

OR J ORourke Art Gal lery Theorems and Algorithms Oxford University Press

T WT Trotter Combinatorics and Partial ly Ordered Sets Dimension Theory The Johns

Hopkins University Press

Circle Packing Representations in Polynomial Time

Bo jan Mohar

The AndreevKo eb eThurston circle packing theorem is generalized and improved

in two ways Simultaneous circle packing representations of the map and its dual map are

obtained such that any two edges dual to each other cross at the right angle The necessary

and sucient condition for a map to have such a primaldual circle packing representation is

that its universal cover graph is connected A p olynomial time algorithm is obtained that

given such a map M and a rational numb er nds an approximation for the primaldual

circle packing representation of M In particular there is a p olynomial time algorithm that

pro duces simultaneous geo desic line convex drawings of a given map and its dual in a surface

with constant curvature so that only edges dual to each other cross

In combination with graph emb edding algorithms these results give rise to ecient algo

rithms for drawing graphs on general surfaces

References

E M Andreev On convex p olyhedra in Lobacevskii spaces Mat Sb N S

Engl transl in Math USSR Sb

E M Andreev On convex p olyhedra of nite volume in Lobacevskii space Mat Sb N S

Engl transl in Math USSR Sb

P Ko eb e Kontaktprobleme auf der konformen Abbildung Ber Verh Saechs Akad Wiss

Leipzig MathPhys Kl

B Mohar Generalizing Kuratowski Theorem this workshop

W P Thurston The geometry and top ology of manifolds Princeton Univ Lect Notes

Princeton NJ

Obstructions for Emb edding Extension Problems

Bo jan Mohar Martin Juvan and Joze Marincek

Kuratowskis theorem gives rise to linear time algorithms which for a given graph determine

whether the graph is planar Hop croft and Tarjan Bo oth and Lueker Fraysseix and

Rosenstiehl FR The extensions of original algorithms also pro duce an emb edding or

nd a Kuratowski subgraph

The result of Kuratowski has b een generalized to nonorientable surfaces by Archdeacon

and Huneke and in a much more general setting to arbitrary surfaces by Rob ertson and

Seymour They proved that for every xed surface there is a nite set of obstructions for

emb eddability of graphs in this surface

University of Ljubljana Slovenia

University of Ljubljana

Although the genus problem is NPhard for every xed surface there is a p olynomial

time algorithm which checks if a given graph can b e emb edded in the surface Filotti et al

Rob ertson and Seymour O n algorithm using graph minors O n log n improvement

by B Reed A constructive version in has running time O n

Let K G and supp ose that we are given a cell emb edding of K into a closed surface

The embedding extension problem asks whether it is p ossible to extend the given emb edding of K

to an emb edding of G In our talk we will briey rep ort on our results concerning the emb edding

extension problem We are able to determine in linear time if the given cell emb edding of K

can b e extended to G In case of a p ositive answer such an extension is exhibited Otherwise an

obstruction is given Obstructions are not always small but they can b e completely characterized

This enables us to get

a a relatively short pro of in total approx pages of the Kuratowskis Theorem for

general surfaces and

b a linear time algorithm that for every xed surface S solves the emb eddability problem

in S In case of the p ositive answer the algorithm also exhibits an emb edding in case of the

negative answer we get a small obstruction a subgraph which can not b e emb edded in S and

whose numb er of branches is small

In a linear time algorithm for emb edding graphs in the pro jective plane is given

presents the solution to the emb edding extension problems in a disk or a cylinder In

Mobius band and the pro jective plane obstructions are characterized and linear time algorithms

for their discovery are presented

The essential work is Let B b e a bridge of K in G It is shown that B contains a

nice small up to a small numb er of almost disjoint millip ede s subgraph B such that if K is

cell emb edded in some surface and F is a face of K then B admits exactly the same typ es of

emb eddings in F as B Moreover such a universal obstruction B can b e constructed in linear

time

In a sp ecial case of the emb edding extension problem is solved when we are restricted

for every bridge of K in G to have at most two essentially dierent emb eddings

The pap er contains a linear time algorithm for emb edding graphs in the torus It is

supp orted by two other works This is the other essential step of our work

In the generalization of the Kuratowski Theorem is proved

Finally presents a linear time algorithm for emb edding graphs in a general xed

surface

References

D Archdeacon The complexity of the graph emb edding problem in Topics in Combina

torics and Graph Theory R Bo dendiek and R Henn ed PhysicaVerlag Heidelb erg

D Archdeacon P Huneke A Kuratowski theorem for nonorientable surfaces J Combin

Theory Ser B

K S Bo oth G S Lueker Testing for the consecutive ones prop erty interval graphs and

graph planarity using PQtrees J Comput System Sci

N Chiba T Nishizeki S Ab e T Ozawa A linear algorithm for emb edding planar graphs

using PQtrees J Comput System Sci

O g

I S Filotti G L Miller J Reif On determining the genus of a graph in O v steps

in Pro c th Ann ACM STOC Atlanta Georgia pp

H de Fraysseix P Rosenstiehl A depthrst search characterization of planarity Ann

Discr Math

J E Hop croft R E Tarjan Ecient planarity testing J ACM

M Juvan J Marincek B Mohar Elimination of lo cal bridges submitted

M Juvan J Marincek B Mohar Corner obstructions submitted

M Juvan J Marincek B Mohar Emb edding graphs in the torus in preparation

M Juvan J Marincek B Mohar Ecient algorithm for emb edding graphs in arbitrary

surfaces in preparation

M Juvan B Mohar A linear time algorithm for the restricted emb edding extension

problem in preparation

B Mohar Pro jective planarity in linear time J Algor in press

B Mohar Obstructions for the disk and the cylinder emb edding extension problem sub

mitted

B Mohar Obstructions for the Mobius band emb edding extension problem submitted

B Mohar Universal obstructions for emb edding extension problems submitted

B Mohar Generalizing Kuratowskis Theorem in preparation

N Rob ertson P D Seymour Graph minors VI I I A Kuratowski theorem for general

surfaces J Combin Theory Ser B

N Rob ertson P D Seymour Graph minors XI I I The disjoint paths problem preprint

last revision of February

N Rob ertson P D Seymour Graph minors XXI Graphs with unique linkages preprint

N Rob ertson P D Seymour Graph minors XXI I Irrelevant vertices in linkage problems

preprint

N Rob ertson P D Seymour An outline of a disjoint paths algorithm in Paths Flows

and VLSILayout Springer pp

C Thomassen The graph genus problem is NPcomplete J Algorithms

S G Williamson Depthrst search and Kuratowski subgraphs J ACM

Automorphisms and Genus on Generalised Maps

Antoine Bergey

Various combinatorial data structures were suggested to enco de top ological maps cell

emb edding of a connected graph in a compact connected orientable surface without b oundary

One of them is combinatorial map or simply maps wich are triples C B such that

and are two p ermutations on the set B The cycles of are called the edges of the

map the cycles of the vertices and those of the faces of the map C Thus we can

LaBRI Universite BORDEAUX I TALENCE b ergeylabriub ord eaux fr

dene in a purely combinatorial way the genus of a map We will say that a map is planar

if its genus is equal to

An automorphism on a map C B consists of a p ermutation such and

Let C b e a combinatorial map enco ding a top ological map C We can asso ciate with

each orientationpreserving automorphism of C an automorphism on C Many analogues

of wellknown results in Riemann surfaces theory were proven concerning automorphisms on

combinatorial maps In particular it can b e shown that an automorphism on a planar map

has two and only two xed p oints

Lienhard suggested nmaps and nGmaps a set B together with n or n p ermutations

on B as extensions of combinatorial maps in order to describ e sub divisions of orientable or

nonorientable spaces of dimension n p ossibly with b oundary

G B is a nGmap if p ermutations i n and pro ducts ji j j

n i i j

are involutions without xed p oints is an involution G is connected

A connected comp onent of G B is called an icell

i i i n

Let us dene an automorphism on a nGmap or nmap G in the following way a p er

mutation wich commutes with every p ermutation of G is called an automorphism of G We

consider automorphisms on nGmaps without b oundary

We show that if an nGmap G is orientable then the automorphisms of G fall into two

of automorphisms orientationpreserving automorphisms and orientation classes and

reversing automorphisms Obviously is a subgroup of the group of the automorphisms of

G b e the nmap asso ciated with an orientable nGmap G As nmaps describ e only Let

orientable sub divisions we can asso ciate in a onetoone manner an orientationpreserving au

tomorphism of G to each automorphism of G

Therefore in dimensions more information ab out the symmetry of a top ological map is

given using Gmaps instead of maps enco ding

We can express any orientationreversing symmetry of a top ological map C as an automor

phism of the asso ciated Gmap G For example when C is a planar map automorphisms

of G are asso ciated with symmetries around the center of the sphere or equatorial planes

As we can use a Gmap G to enco de graph drawings on nonorientable surfaces we can

also treat symmetries of such emb eddings as an automorphism of G

Let b e an automorphism of a Gmap G We show that there are two classes of cells

that are xed by let us call them even and o ddcells resp ectively Let us consider relations

b etween genus orientability of G and the numb er of cells left xed by

When G is orientable has only o ddcells if it is an orientationreversing automorphism even

cells if it preserves the orientation

If the genus of G is equal to then has two evencells if G is orientable one if G is not

orientable

We show that every o ddcell is adjacent exactly to two o ddcells Thus we can dene ovals of

o ddcells These adjacency relations implie that

each oval contains an even numb er of o ddcells

we have n n n n n n n n n where n n and n are the numb ers

v f e e f v f v e e v f

of o ddedges o ddfaces and o ddvertices resp ectively

If there are at least one o ddcell then the automorphism is an involution When G is orientable

these ovals can b e considered as the intersection of the symmetry plane asso ciated with with

the surface on which the graph is drawn Moreover when G is orientable we nd an analogue

to Harnacks theorem More precisely we have

g n

where g is the genus of G and n the numb er of ovals This is shown by exhibiting G a

Gsubmap of G wich is of genus n

Let C b e a planar top ological map C enco ded by a Gmap G and let O b e an oval of an

automorphism on G Splitting the top ological map along O let us get symmetrical drawings in

the following way we divide C along the cells of O into two parts C and C we draw C on

the topside of the sphere each cell of O lying on the equatorial plane then the b ottomside

of the sphere is symmetrical in relation to the equatorial plane

In other algorithms these splittings may b e usefull in order to save space or time

References

R Cori A Mach Maps and Hyp ermaps a survey I I I I I I Exp ositiones Mathematicae

R Cori A Mach JG Penaud and B Vauquelin On the automorphism group of a planar

hyp ermap Europ J Combinatorics

J Edmonds A combinatorial representation for p olyhedral surfaces Notices Amer Math

So c

P Lienhardt Top ologic mo dels for b oundary representation a comparison with n

dimensional generalized maps computeraided design

P Lienhardt Sub divisions de surfaces et cartes generalisees en dimension RAIRO The

oretical Informatics and Appications

Upward Drawing on Surfaces

Ivan Rival

Extended Abstract

The mo dern theoretical computer science literature is preo ccupied with ecient data structures

to co de and store ordered sets Among these data structures graphical ones play a decisive

role esp ecially in decisionmaking problems Choices must b e made from among alternatives

ranked hierarchically according to precedence or preference And lo osely sp eaking graphical

data structures must b e drawn in order that they may b e easily read

Besides the well known metaphors inspired by layout design project management and

database design several unexp ected application areas are driving our recent investigations

Department of Computer Science University of Ottawa Ottawa KN N Canada rivalcsiuottawaca

i Ice ows consisting of vast areas of ice largely of recent vintage a few years old inter

sp ersed with old ice many years old p ose a profound danger to b oats and oilrigs indeed

for any manmade vessel at all Ocean currents wind and temp erature aect the iceb ergs

direction of ow changing p osition and velo city substantially even within hours

ii The increasing use of p ersonal workstations has led to program visualization techniques in

order to grasp complex computer programs Fisheye techniques provide one such to ol to

elucidate the structure and b ehaviour of computer programs That in turn can simplify

and hence advance the eectiveness of programming

iii Inspired by the problem to unify the known forces quantum topology combines space and

time to pro duce a dimensional picture of the world In this raried air current research

in physics meets up with classical mathematics

From the viewp oint of graph drawing there is a common thread to these themes Each

involves an ordered set whether it consists of moving iceb ergs hierarchies of subroutines or

light cones and each views the ordered set monotonical ly on a twodimensional surface We

are led ineluctably to study upward drawings of ordered sets with vertices drawn on an oriented

surface usually in space whose edges are monotonice paths with resp ect to the zaxis The

to ols of topological graph theory and the traditional machinery of dierential topology may b e

brought to b ear

Here are some of the highlights of our work

Every trianglefree graph has a planar upward drawingKisielewiczRival

Every ordered set has an upward drawing on a vertical multipleholed torus

NowakowskiRival

The order genus of an ordered set is an invariant among al l of the orientations of its

covering graphEwachaLiRival

Every bounded ordered set of genus g has an upward drawing on a surface with precisely g

sadd lepoints MusinRivalTarasov

These results in turn create new algorithmic considerations For instance as the covering

graph of an ordered is trianglefree we must replace the commonplace reliance on triangula

tion by an analogous but dierent sub division pentangulation They inspire intriguing and

disarmingly simple questions to o does every ordered set have a cel lular upward drawing on a

surface

References

K Ewacha W Li and I Rival Order genus and diagram invariance ORDER

A Kisielewicz and I Rival Every trianglefree graph has an upward drawing ORDER

O R Musin I Rival and S Tarasov Upward drawings on surfaces and the index of

singularity manuscript

R Nowakowski and I Rival Bending and stretching orders into three channels Technical

report University of Ottawa

I Rival Reading drawing and order Algebras and Order eds I Rosenb erg and G

Sabidussi NATO ASI Kluwer pp

Tessalation and Visibility Representations of Maps on the

Torus

Bo jan Mohar and Pierre Rosenstiehl

It is shown that maps on the torus whose universal covering graph is connected b ehave

very much like connected plane graphs In particular the results on visibility tessalation

representations and upward drawings of plane graphs are generalized to maps on the torus

A Simple Construction of High Representativity

Triangulations

Teresa M Przytycka and Jozef H Przytycki

One natural way of drawing a top ological surface is to start with a drawing of its triangu

lation Thus one can ask which triangulation of a given surface leads to a nice drawing of the

surface The b est candidate for such a triangulation is a triangulation in which all vertices are

evenly distributed over the surface Such a triangulation can b e achieved by maximizing a

parameter of surface triangulation called representativity

The concept of the representativity of a graph emb edding was intro duced by Rob ertson and

Seymour Rob ertson and Vitray consider as a ma jor eect of high representativity the fact

that it makes the emb edding highly lo cally planar and that the lo cally Euclidean prop erty

of the surface is mirrored by the lo cally planar prop erty of the emb edded graph

Formally the representativity of a graph G emb edded in a surface is equal to the length

of the shortest noncontractible facial walk a walk of typ e v f v f v f v where for

k k

any i v is a vertex f is a face of the graph and v v are vertices of f In particular the

i i i i i

representativity of a surface triangulation is equal to the length of the shortest noncontractible

cycle of the triangulation Recall that a cycle C on a surface is called noncontractible if

none of the comp onents of C is homeomorphic to an op en disc

It is not known what is the highest p ossible representativity that can b e achieved when tri

angulating a genus g surface with an nvertex graph Joan Hutchinson showed that the

representativity of such a triangulation is at most c ng log g where c is a constant Hutchin

son gave a simple construction that allows to triangulate with representativity ng This

p p

ng log g and further ng log log g using a covering result was improved to

spaces technique

University of Ljubljana Slovenia and EHESS Paris

Department of Mathematics and Computer Science Odense University DK Odense M Denmark przy

tyckimadaoudk jozefimadaoudk

In this pap er we improve substantially the previously known lower b ound for the repre

sentativity of such a triangulation and give an ecient algorithm for its construction More

precisely we present an O maxg ntime algorithm that for given g and n such that g

and n c g log g constructs an nvertex triangulation of a genus g surface such that the

ng log g where c c are constants We representativity of the triangulation is at least c

extend our result to nonorientable surfaces and surfaces with b oundary In the later case we

replace the genus g of the surface with parameter g g d where d is the numb er of

b oundary comp onents

In our construction we use a relation b etween cubic graphs and orientable surfaces without

b oundary Namely given a N vertex cubic graph G one can construct an orientable surface

where g N In fact can b e taken as a tubular neighb orho o d of G emb edded in R

g g

Then we can move G so that it is emb edded in and thus all cycles of G are noncontractible

Let b e the girth the length of the shortest cycle of G Given the emb edding of a cubic

graph G as describ ed ab ove we construct a triangulation of by adding new vertices in

such a way that no noncontractible cycle shorter than is intro duced The total numb er of

vertices added is N O N The construction takes O N time Thus to maximize the

g g

representativity of the triangulation obtained using this construction we should start with G

b eing a cage where a cage is a cubic graph with girth and the smallest p ossible

numb er of vertices Unfortunately there is no ecient algorithm that for a given value

constructs a cage However by a theorem of Erdos and Sachs for any integer there

exists a cubic graph of girth and O vertices

Based on the pro of of the theorem of Erdos and Sachs we give an O N time algorithm to

construct a cubic graph with N vertices and girth log N This gives us for any N

an N vertex triangulation T of such that representativity of T is log N and N

g g g g g

O N log N Thus the lower b ound claimed is achieved for n N The next step is to extend

the lower b ound to all values of n We achieve this by appropriate sub division of the triangulation

T Finally we show an extension of the construction to nonorientable surfaces and surfaces

with b oundary We achieve this by cutting some handles of the surface along edges of the

dg e

T To obtain triangulations of nonorientable surfaces we cap o at most two of the holes

dg e

created in the ab ove construction with triangulated Mobius bands

References

PErdos HSachs Regulare Graphen gegeb ener Taillenweite mit minimaler Knotenzahl

Wiss Z Univ HalleWittenb erg MathNat

JHutchinson On short noncontractible cycles in emb edded graphs SIAM J Disc Math

TMPrzytycka JHPrzytycki On lower b ound for short noncontractible cycles in emb ed

ded graphs SIAM J Disc Math

TMPrzytycka JHPrzytycki Surface Triangulations Without Short Noncontractible Cy

cles Contemporary Mathematics Graph Structure Theory

NRob ertson PDSeymour Graph minors VI I Disjoint paths on a surface JComb The

ory Ser B

NRob ertson RVitray Representativity of surface emb eddings Algorithms and Combi

natorics Volume Paths Flows and VLSILayout SpringerVerlag Berlin Heidelb erg

Volume Editors B Korte L Lovasz H J Promel and A Schrijver

On a Visibility Representation for Graphs in Three

Dimensions

y z x

Prosenjit Bose Hazel Everett Sandor Fekete Anna Lubiw Henk Meijer

k yy

Kathleen Romanik Tom Shermer and Sue Whitesides

Visibility representations of graphs map vertices to sets in Euclidean space and express

edges as visibility relations b etween these sets Application areas such as VLSI wire routing

and circuit b oard layout have stimulated research on visibility representations where the

sets b elong to R Here motivated by the emerging research area of graph drawing we

study a dimensional visibili ty representation

Consider an arrangement of closed disjoint rectangles in R such that the planes deter

mined by the rectangles are p erp endicul ar to the z vertical direction and the sides of the

rectangles are parallel to the x or y horizontal directions A vertical thick line of sight

b etween two rectangles R and R is a closed cylinder C of nonzero length and radius such

i j

that the ends of C are contained in R and R the axis of C is parallel to the z direction

i j

and the intersection of C with any other rectangle in the arrangement is empty We call

such an arrangement a Brepresentation of an abstract graph G V E if and only if

the following hold

there exists a onto corresp ondence b etween the rectangles and the vertices and

vertices v and v are adjacent if and only if their corresp onding rectangles R and

i j i

R have a vertical thick line of sight b etween them

Our main results are as follows All planar graphs are Brepresentable as are many non

planar graphs In particular K is Brepresentable for all m and n and K is B

mn n

representable for values of n However K is not Brepresentable for n We

have also considered variants of Brepresentations

Brepresentability of planar graphs The pro of that all planar graphs are Brepresentable

has two main ingredients The rst is the result due indep endently to WismathW and

to Tamassia and TollisTT that any connected planar graph has what TT calls an

visibility representation Vertices corresp ond to closed disjoint horizontal line segments

in the plane and two vertices are adjacent in the graph if and only if their corresp onding

segments can b e joined by a vertical band of nonzero width and length with ends lying

in the segments The second ingredient is the use of the rd dimension to deal with cut

vertices This is similar to an idea of W for obtaining a visibili ty representation for all

planar graphs by rectangles in R that can lo ok in b oth x and y directions

McGill University Canada jitmucsmcgillca

Universite du Queb eca Montreal Canada hazelopuscsmcgill ca

SUNY Stony Bro ok USA sandoramssunysbedu

University of Waterlo o Canada alubiwmaytaguwaterlo oca

Queens University Canada henkqucisqueensuca

McGill University Canada romanikopuscsmcgill ca

Simon Fraser University Canada shermercssfuedu

McGill University Canada suecsmcgillca

Brepresentability of K and K K has a simple general Brepresentation for all

mn n mn

m n but this is not the case for K While we have constructed an explicit Brepresentation

for K and hence for K n we have also shown that K has no Brepresentation

n n

for n We conjecture that n is close to the correct b ound

NonBrepresentability of K for n In any Brepresentation of a complete graph

no two rectangles can lie on the same z constant plane Also if K has a Brepresentation

then it has one in which no two rectangles have sides at the same x or y constant values

In other words K can b e represented by rectangles that can b e linearly ordered by z

co ordinate and also by x co ordinate of left side x co ordinate of right side y co ordinate

of top side and y co ordinate of b ottom side To obtain the result we use such orderings

together with rep eated application of the following result of Erdos and SzekeresES

For any positive integers j and k any sequence of more than j k distinct integers has a not

necessarily contiguous increasing subsequence of length j or decreasing subsequence of

length k

Variations Our representation of K can b e carried out with squares provided the squares

need not have the same area Representations by discs unit squares and squares for com

plete graphs and for complete bipartite graphs have b een considered

Acknowledgment Our study of Brepresentations b egan at Bellairs Research Institute

of McGill University during the Workshop on Visibility Representations organized by S

Whitesides and J Hutchinson February We are grateful to the other conference

participants Joan Hutchinson Go os Kant Marc van Kreveld Bepp e Liotta Steve Skiena

Rob erto Tamassia Yanni Tollis and Go dfried Toussaint

References

ES Erdos P and Szekeres A A combinatorial problem in geometry Compositio

Mathematica v

TT Tamassia R and Tollis I G A unied approach to visibili ty representations of

planar graphs Discrete Comput Geom v

W Wismath Stephen Kenneth BarRepresentable Visibili ty Graphs and a Related Net

work Flow Problem University of British Columbia Dept of Computer Science Tech

nical Rep ort August

On Graph Drawings with Smallest Numb er of Faces

Jianer Chen Saro ja P Kanchi and Jonathan L Gross

We rep ort here our recent progress in the study of graph drawings with the smallest numb er

of faces or equivalently graph emb eddings with the largest genus Formally the maximum

Department of Computer Science Texas AM University College Station TX chencstamuedu

skanchicstamuedu

Department of Computer Science Columbia University New York NY

genus G of a connected graph G is dened to b e the largest integer k such that there

exists a cellular emb edding of G into the orientable surface of genus k

Since the intro ductory investigation of maximum genus by Nordhaus Stewart and White

there has b een considerable interest in maximum genus emb eddings of graphs Two out

standing results in this research are Xuongs characterization of maximum genus emb edding

in terms of comp onents of the complements of spanning trees and a rst p olynomialtime

algorithm for computing maximum genus develop ed by Furst Gross and McGeo ch

Recent investigations have fo cused on deriving a lower b ound on the maximum genus of

graphs Skoviera showed that the maximum genus of a connected graph of diameter

is at least d Ge More recently Chen and Gross proved that the maximum genus

of a connected simplicial graph or of a connected graph is at least log G The last

result was further improved by Chen Gross and Riep er who proved that the maximum

genus of a connected simplicial graph G is at least G A related topic the upp er

emb eddability of graphs has also b een studied extensively in literature

In this pap er we prove that G G for a simplicial graph G and we show that

our b ound is tight Our pro of selectively sharp ens Xuongs characterization of the maximum

genus emb edding We rst show that every regular simplicial graph G has a Xuong tree T

such that every o dd comp onent in the Xuong cotree G T has only one edge This enables

us to compare the numb er of o dd comp onents to the numb er of even comp onents in the

Xuong cotree and thereby arrive at an upp er b ound for the numb er of o dd comp onents

This upp er b ound is used to obtain the desired lower b ound for the maximum genus of

a regular simplicial graph Finally the restriction of regularity is removed by using a

theorem of Chen and Gross concerning edgecontractions

Our result on the lower b ound of maximum genus has several interesting consequences to

the average genus of graphs Using techniques develop ed by Chen and Gross we show that

the average genus of a simplicial graph is at least of its cycle rank This improves a

result by Chen Gross and Riep er that the average genus of a connected simplicial

graph is at least of its cycle rank Moreover our result implies that the average genus

of a simplicial graph is at least of its maximum genus thereby complementing another

result of Chen Gross and Riep er that for a regular graph the average genus is at least

half its maximum genus

References

J Chen and J L Gross Limit p oints for average genus I connected and

connected simplicial graphs J Combinatorial Theory B

J Chen J L Gross and R G Rieper Lower b ounds for the average genus

Submitted for publication

M L Furst J L Gross and L A McGeoch Finding a maximumgenus

graph imb edding J of ACM

L Nebesky Every connected lo cally connected graph is upp er imb eddable J

Graph Theory

E Nordhaus B Stewart and A White On the maximum genus of a graph

J Combinatorial Theory B

R Ringeisen The maximum genus of a graph PhD thesis Department of

Mathematics Michigan State University East Lansing MI

R Ringeisen Determining all compact orientable manifolds up on which K

has cell imb eddings J Combinatorial Theory B

M Skoviera The maximum genus of graphs of diameter two Discrete Mathematics

N H Xuong How to determine the maximum genus of a graph J Combinatorial

Theory B

N H Xuong Upp erimb eddable graphs and related topics J Combinatorial Theory

J Zaks The maximum genus of cartesian pro ducts of graphs Canad J Math

A Flow Mo del of Low Complexity for Twisting a Layout

Marc Bousset

We deal with s tbip olar orientations of a connected plane graph G taking lo cal orien

tation constraints into account A laterality constraint links the orientations of two edges

adjacent in the circular order around a vertex in the following way

an extremal constraint on an angle implies that the two edges adjacent to that angle

are either b oth outgoing or b oth incoming

a lateral constraint on an angle implies that among the two edges adjacent to that

angle one is incoming and one is outgoing

Our approach is based on the theory of the s tbip olar marking of the angle graph G a

sp ecial colour marking of the edges of G intro duced by P Rosenstiehl There is a one

toone corresp ondance b etween s tbip olar markings of G and s tbip olar orientations

of G

The lo cal invariants on the colours around each vertex and around each face allow us to

translate the problem into mathematical programming namely into a ow mo del The

network is built up on the angle graph G to which we add two no des S and T We also add

edges from S to each vertex of G Vedges and from each face of G to T Fedges The

capacities of the angles are when there are no constraints The capacities of the Vedges

are except for S s and S t for which they are zero The capacities of each Fedge f T

is equal to the degree of the face f minus A maximum ow in this network saturates

b oth the Vedges and the Fedges and the ow through each angle gives an s tbip olar

marking

The integer capacities around S and T and the unit capacities elsewhere allow us to extend

a result of R E Tarjan that is a drastic reduction of the complexity of the maximum

ow algorithm down to O m m for Dinics algorithm

Laterality constraints are enforced by mo difying the capacities on the angles When no

s tbip olar marking consistant with the constraints exists the maximum ow do es not

CAMS EHESS Paris b oussetdassaultavi on fr

saturate all the Vedges and the Fedges It is then p ossible to identify a set of angles I so

that relaxing the constraints on all angles of I guarantees the existence of an s tbip olar

marking compatible with the remaining constraints

The metho d develop ed here has b een implemented in an industrial context in a CAD system

for generating layouts of electrical networks and also in the TWIST software created for

the ALCOM pro ject

References

M Bousset and P Rosenstiehl TWIST Version ALCOM rep ort

R Cori Un code pour les graphes planaires et ses applications volume of Asterisque

So ciete Mathematique de France

H de Fraysseix P Ossona de Mendez and P Rosenstiehl Bip olar orientations revis

ited Discrete and Applied Mathematics to app ear

E A Dinic Algorithm for solution of a problem of maximum ow in a network with

p ower estimation Soviet Math Dokl

S Even and R E Tarjan Network ow and testing graph connectivity SIAM J

Comput

P Rosenstiehl Emb edding in the plane with orientation constraints The angle graph

In Proceedings of the Third International Conference New York pages

Annals of the New York Academy of Sciences

P Rosenstiehl and R E Tarjan Rectilinear planar layouts and bip olar orientations

of planar graphs Discrete and Computational Geometry

R E Tarjan Data Structures and Network Algorithms So ciety for Industrial and

Applied Mathematics

M Veldhorst A bibliography on network ow problems Algorithms Review

ALCOM

Convex and nonConvex Cost Functions

of Orthogonal Representations

y z x

Giusepp e Di Battista Giusepp e Liotta and Francesco Vargiu

Work partially supp orted by Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo of the Italian Na

tional Research Council CNR and by Esprit BRA of the EC Under Contract Alcom I I

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza via Salaria I Roma

Italia dibattistaiasirmcnri t

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza via Salaria I Roma

Italia Part of this work has b een done when this author was visiting the Department of Computer Science of

McGill University liottainfokiti nguni roma it

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza via Salaria I Roma

Italia and Database Informatica Spa vargiuinfokiti nguni roma it

An orthogonal drawing of a graph is a planar drawing such that all the edges are p olygonal

chains of horizontal and vertical segments Finding the planar emb edding of a planar graph

such that its orthogonal drawing has the minimum numb er of b ends is a fundamental op en

problem in graph drawing We provide the rst partial solution to the problem First we give

a new combinatorial characterization of orthogonal representations of connected graphs

based on the concept of spirality and we relate the numb er of b ends of representations to the

spirality by means of the concept of cost function Second we exploit the b ehaviour of cost

functions asso ciated to orthogonal representations of comp onents of connected graphs

Third we use the characterization to nd in p olynomial time the planar emb edding of a

seriesparallel graph and of a connected planar graph such that its orthogonal drawing

has the minimum numb er of b ends

In this talk we give a new combinatorial characterization of orthogonal representations of

connected graphs and use the results to provide a rst partial solution to the problem of

nding the planar emb edding of a planar graph such that its of orthogonal drawing has the

minimum numb er of b ends

First we intro duce the new concept of spirality that is a measure of how an orthogonal

drawing is rolled up giving a combinatorial characterization that relates the numb er of

b ends of an orthogonal drawing and its spirality by means of the concept of cost function

Second we study the b ehaviour of cost functions of connected graphs with the following

results

Cost functions of comp onents of planar graphs are non decreasing convex piecewice

linear functions

Cost functions of comp onents of planar graphs are piecewice linear functions but

p ossibly concave wshap e

Third from the ab ove results we solve in p olynomial time the problem of nding the planar

emb edding that leads to the orthogonal drawing with the minimum numb er of b ends for

planar graphs and seriesparallel graphs

Minimizing the numb er of b ends of orthogonal representations is a classical problem of

graph drawing

Tamassia has prop osed a very elegant representation algorithm that solves the prob

lem in p olynomial time for graphs with a xed emb edding The algorithm is based on a

combinatorial characterization that allows to map the problem into a mincostow one

Linear time heuristics for the same problem have b een prop osed by Tamassia and Tollis

in Such heuristics guarantee at most n b ends for a biconnected graph with n

vertices Recently Kant has prop osed ecient heuristics with b etter b ounds for tricon

nected planar graphs and general planar graphs a graph is k planar if it is planar and

each vertex has degree at most k Tamassia Tollis and Vitter have given lower b ounds

for the problem and the rst parallel algorithm A brief survey on orthogonal drawings is

However all the ab ove pap ers work within a xed emb edding where it can b e seen that

the choice of the emb edding can deeply aect the results obtained by the algorithms The

problem of nding the planar emb edding that leads to the minimum numb er of b ends is

not known to b e NPhard or not and has b een explicitely mentioned as op en by several

authors Although the problem is quite natural there are only a few contributions on this

topic b ecause of the exp onential numb er of emb eddings a planar graph in general has

Our technique exploits b oth the prop erties of the spirality and a variation of the S P QR

trees a data structure that implicitely represents all the planar emb eddings of a

planar graph Moreover we adopt a slight mo dication of the algorithm of Tamassia for

computing orthogonal representation of triconnected comp onents

Observe that seriesparallel graphs arise in a variety of problems such as scheduling elec

trical networks dataow analysis database logic programs and circuit layout Also they

play a very sp ecial role in planarity problems

References

P Bertolazzi RF Cohen G Di Battista R Tamassia and IG Tollis How to Draw a

SeriesParallel Digraph Pro c rd Scandinavian Workshop on Algorithm Theory

G Di Battista and R Tamassia Incremental Planarity Testing Pro c th IEEE Symp on

Foundations of Computer Sciene pp

G Di Battista and R Tamassia On Line Planarity Testing Technical Rep ort CS

Dept of Computer Science Brown Univ

P Eades and R Tamassia Algorithms for Automatic Graph Drawing An Annotated Bib

liography Technical Rep ort CS Dept of Computer Science Brown Univ

S Even Graph Algoritms Computer Science Press Potomac MD

D R Fulkerson An OutofKilter Metho d for Minimal Cost Flow Problems SIAM Journal

Appl Math no pp

G Kant A New Metho d for Planar Graph Drawings on a Grid Pro c IEEE Symp on

Foundations of Computer Science

E L Lawler Combinatorial Optimization Networks and Matroids Holt Rinehart and

Winston New York Chapt

R Tamassia On Emb edding a Graph in the Grid with the Minimum Numb er of Bends

SIAM J Computing vol no pp

R Tamassia Planar Orthogonal Drawings of Graphs Pro c IEEE Int Symp on Circuits

and Systems

R Tamassia G Di Battista and C Batini Automatic Graph Drawing and Readability of

Diagrams IEEE Transactions on Systems Man and Cyb ernetics vol SMC no pp

R Tamassia and IG Tollis Ecient Emb edding of Planar Graphs in Linear Time Pro c

IEEE Int Symp on Circuits and Systems Philadelphia pp

R Tamassia and IG Tollis Planar Grid Emb edding in Linear Time IEEE Trans on

Circuits and Systems vol CAS no pp

R Tamassia I G Tollis and J S Vitter Lower Bounds and Parallel Algorithms for Planar

Orthogonal Grid Drawings Pro c IEEE Symp on Parallel and Distributed Pro cessing

K Takamizawa T Nishizeki and N Saito LinearTime Computability of Combinatorial

Problems on SeriesParallel Graphs Journal of the ACM vol no pp

J Valdes RE Tarjan and EL Lawler The Recognition of Series Parallel Digraphs SIAM

J on Comp No

L Valiant Universality Considerations in VLSI Circuits IEEE Trans on Computers vol

c no pp

Top ology and Geometry of Planar Triangular Graphs

y z

Giusepp e Di Battista and Luca Vismara

The contribution of this talk is twofold On one side we give several top ological results

a new op eration to construct plane triangular graphs from a triangle graph a

basic lemma ab out the planarity of all the drawings of plane triangular graphs a

new ordering for the vertices of plane triangular graphs On the other side we give a

characterization of all the planar drawings of a triangular graph by means of a system of

equations and inequalities relating its angles solving a problem that is explicitel y mentioned

as op en by several authors we also discuss minimality prop erties of the characterization

The characterization can b e used to decide in linear time whether a given distribution

of angles b etween the edges of a planar triangular graph can result in a planar drawing

to tackle the problem of maximizing the minimum angle of the drawing of a planar

triangular graph by studying the solutionspace of a nonlinear optimization problem

to give a characterization of the planar drawings of a triconnected graph through a system

of equations and inequalities relating its angles to give a characterization of Delaunay

triangulations through a system of equations and inequalities relating its angles to give

a characterization of all the planar drawings of a triangular graph through a system of

equations and inequalities relating the length of its edges in turn this result allows to give

a new characterization of the disc packing representations of planar triangular graphs

Intro duction

Planar straightline drawings of planar graphs are a classical topic of the graph drawing

eld surveys on graph drawing can b e found in

A classical result established by Wagner Fary Stein and Steinitz shows that every planar

graph has a planar straightline drawing

A grid drawing is a drawing in which the vertices have integer co ordinates Indep endently

de Fraysseix Pach and Pollak and Schnyder have shown that every nvertex

planar graph has a planar straightline grid drawing with O n area

Straightline drawings have also b een studied with the constraint for all the faces to b e

represented by convex p olygons convex drawings Tutte shows that for a tricon

nected graph convex drawings can b e constructed by solving a system of linear equations

Recently Kant has shown an algorithm to construct grid convex drawings with quadratic

area

In the research on planar straightline drawings a very sp ecial role is played by angles

b etween the segments that comp ose the drawing In particular Vijayan studied angle

graphs An angle graph is a planar emb edded graph in which the angles b etween successive

edges incident at vertices are given The problem of the existence of a planar straightline

Research supp orted in part by ESPRIT Basic Research Action No ALCOM I I by Progetto Finalizzato

Sistemi Informatici e Calcolo Parallelo of the Italian National Research Council CNR

Dipartimento di Informatica e Sistemistica Universita di Roma La Sapienza Via Salaria

Roma Italy dibattistaiasi rmcnrit

Research p erformed in part while this author was at IASICNR Dipartimento di Informatica e Sistemistica

Universita di Roma La Sapienza Via Salaria Roma Italy vismaraiasirmcnrit

drawing of an angle graph that preserves the angles is tackled and partial characterization

results are shown

In the problem of constructing straightline drawings of graphs with large angles is

studied It is shown that it is always p ossible to construct a drawing whose smallest angle

b etween the edges incident at a vertex is O d where d is the maximum vertex degree

of the graph Other results are given for particular classes of graphs For planar graphs the

b ound is improved to O d however in general the obtained drawing is nonplanar

Malitz and Papakostas have shown that it is always p ossible to construct a planar

straightline drawing of a planar graph whose smallest angle is O where

The b ound that is presented is only existential in fact they exploits a disc packing

representation of the graph In a disc packing representation each vertex is a disc two

vertices are adjacent in the graph if and only if their discs are tangent and the interiors

of the discs are pairwise disjoint No p olynomial algorithm is known to construct a disc

packing representation Recently Mohar has shown for this problem an approximation

algorithm

A fundamental to ol for several algorithms and characterizations describ ed ab ove are planar

triangular graphs For instance the algorithm by de Fraysseix et al and the algorithm

by Schnyder have an intermediate step in which the given planar graph is triangulated

Also planar triangular graphs play a very sp ecial role in a numb er of problems arising

in Computational Geometry However as far as we know characterizing angles of planar

triangular graphs has b een an elusive goal for a long time The contribution of this pap er

can b e summarized as follows

We dene a new op eration named closewheel such that any plane triangular graph

G with n vertices can b e constructed starting from the triangle graph by a sequence

of O n closewheel op erations

We prove a lemma ab out the planarity of all the drawings of plane triangular graphs

We dene a new ordering metho d for the vertices of plane triangular graphs

We give a characterization of all the planar drawings of a triangular graph through

a system of equations and inequalities relating its angles The problem is explicitely

mentioned as op en by several authors see eg We also discuss minimality

prop erties of the characterization

The characterization ab ove has several applications

It can b e used to decide in linear time whether a given distribution of angles b etween

the edges of a planar triangular graph can result in a planar drawing

It allows to tackle the problem of maximizing the minimum angle of the drawing of

a planar triangular graph by studying the solutionspace of a nonlinear optimization

problem

It gives a characterization of the planar drawings of a triconnected graph through a

system of equations and inequalities relating its angles

It gives a characterization of Delaunay triangulations through a system of equations

and inequalities relating its angles solving a problem stated in Recently the prob

lem of characterizing angles of Delaunay triangulations has b een tackled by Dillencourt

and Rivin who have shown that a system of equations and inequalities relating the

angles of a plane triangular graph G can b e used to decide whether G can b e drawn

as a Delaunay triangulation

It can b e exploited to give a characterization of all the planar drawings of a trian

gular graph through a system of equations and inequalities relating the length of its

edges in turn this result allows to give a new characterization of the disc packing

representations of planar triangular graphs

References

FJ Brandenburg P Kleinschmidt and U Schnieders Drawing Planar Graphs with

Wide Angles Manuscript

H de Fraysseix J Pach and R Pollack Small Sets Supp orting Fary Emb eddings

of Planar Graphs Pro c th ACM Symp on Theory of Computing pp

H de Fraysseix J Pach and R Pollack How to Draw a Planar Graph on a Grid

Combinatorica vol pp

MB Dillencourt GraphTheoretical Prop erties of Algorithms Involving Delaunay

Triangulations Tech Rep CSTR University of Maryland

MB Dillencourt Toughness and Delaunay Triangulations Discrete Computa

tional Geometry vol no pp

MB Dillencourt and I Rivin Personal Communication

P Eades and R Tamassia Algorithms for Automatic Graph Drawing An Anno

tated Bibliography Technical Rep ort CS Dept of Computer Science Brown

Univ

I Fary On Straight Lines Representation of Planar Graphs Acta Sci Math

Szeged vol pp

M Formann T Hagerup J Haralambides M Kaufmann FT Leighton A Simvo

nis E Welzl and G Wo eginger Drawing Graphs in the Plane with High Resolu

tion Pro c IEEE Symp on Foundations of Computer Science pp

CD Ho dgson I Rivin and WD Smith A Characterization of Convex Hyp erb olic

Polyhedra and o d Convex Polyhedra Inscrib ed in the Sphere Bull of the American

Mathematical So ciety vol no pp

G Kant A New Metho d for Planar Graph Drawings on a Grid Pro c IEEE Symp

on Foundations of Computer Science

S Malitz and A Papakostas On the Angular Resolution of Planar Graphs Pro c

th ACM Symp on the Theory of Computing pp

B Mohar Circle packings of maps in p olynomial time Manuscript submitted to

the Bull of the American Mathematical So ciety

T Nishizeki and N Chiba Planar Graphs Theory and Algorithms Annals of Dis

crete Mathematics North Holland

W Schnyder Emb edding Planar Graphs on the Grid Pro c ACM SIAM Symp

on Discrete Algorithms pp

SK Stein Convex Maps Pro c Amer Math So c vol pp

E Steinitz and H Rademacher Vorlesung ub er die

R Tamassia G Di Battista and C Batini Automatic Graph Drawing and Read

ability of Diagrams IEEE Transactions on Systems Man and Cyb ernetics vol

SMC no pp

WT Tutte Convex Representations of Graphs Pro c London Math So c vol

WT Tutte How to Draw a Graph Pro c London Math So c vol no pp

G Vijayan Geometry of Planar Graphs with Angles Pro c ACM Symp on Com

putational Geometry pp

K Wagner Bemerkungen zum Vierfarb enproblem Jb er Deutsch MathVerein

vol pp

An Optimal PRAM Algorithms for Planar Convex

Emb edding

y z x

Frank Dehne Hristo Djidjev and JorgRudiger Sack

Intro duction

The task of representing a diagram in understandable and readable form arises in a variety

of areas including eg circuit design and information system analysisdesign The reader

is referred to Eades and Tamassia for an annotated bibliography of graph drawing

algorithms

The existence of a planar representation of a graph can b e tested in linear time as was

rst proved by Hop croft and Tarjan Chiba et al provided a linear time algorithm

to nd a planar representation of a planar graph Their result is based on an algorithm to

determine the existence of a planar representation due to Bo oth and Lueker

A sub class of planar graphs whose representation is particularly aestetically pleasing are

those planar graphs whose b ounded faces can all b e drawn convexly ie the emb edding of

each b ounded face is a convex p olygon Thomassen describ es a sequential metho d for

emb edding a planar graph once an extendible outer cycle F is given An extendible outer

facial cycle F of G can b e determined in linear time as describ ed in See also for

other related work K

Results

We give an optimal parallel algorithm for determining whether a graph can b e convexly

emb edded and if so for constructing such a convex emb edding for the graph The algorithm

develop ed runs in O log n time with O n space and the same pro cessor b ound as graph

connectivity

This research was partially supp orted by Natural Sciences and Engineering Council of Canada

Scho ol of Computer Science Carleton University Ottawa Ontario KS B Canada dehnescscarletonca

Department of Computer Science Rice University Houston Texas USA

Scho ol of Computer Science Carleton University Ottawa Ontario KS B Canada sackscscarletonca

Our solution to this problem is based on the following Using stnumb ers we present a

decomp osition scheme and develop a parallel algorithm to determine such a decomp osition

To emb ed the paths so that each face of the original is convex we generalize the notion of

convexity of p olygons to pseudoconvexity We give an algorithm to emb ed a path inside

a pseudoconvex p olygon so that the resulting subp olygons are also pseudoconvex When

applying this emb edding to the path decomp osition a convex emb edding of the entire graph

is obtained

We use the following parallel algorithms planarity testing st numb ering list

ranking lowest common ancestor in a tree biconnectivity this algorithm is not

optimal as describ ed but with listranking algorithm it b ecomes optimal triconnectivity

parallel transitive closure and p oint lo cation in planar structures

References

O Berkman B Schieb er and U Vishkin Some doubly logarithmic optimal paral lel

algorithms based on nding al l nearest smal ler values Technical Rep ort UMIACSTR

University of Maryland

K Bo oth G Lueker Testing for the consecutive ones property interval graphs and

graph planarity using PQtree algorithm J Comp Syst Sci pp

N Chiba K Onoguchi T Nishizeki Drawing planar graphs nicely Acta Informatica

N Chiba T Yamanouchi T Nishizeki Linear algorithms for convex drawings of

planar graphs in Progress in Graph Theory JA Bondy and USR Murth eds

Academic Press pp

R Cole U Vishkin Optimal paral lel algorithms for expression tree evaluation and list

ranking Pro c AWOC Lecture Notes in Computer Science SpringerVerlag

G Di Battista P Eades R Tamassia and IG Tollis Algorithms for drawing graphs

an annotated bibliography Technical Rep ort Brown University up dated

D Fussell V Ramachandran R Thurimella Finding triconnected components by local

replacements Pro c ICALP LNCS SpringerVerlag pp

H Gazit Optimal EREW paral lel algorithms for connectivity ear decomposition and

stnumbering of planar graphs Pro c th Int Parallel Pro cessing Symp

J Hop croft and RE Tarjan Ecient plarnarity testing JACM pp

G Kant Drawing Planar Graphs Using the lmcOrdering Pro c IEEE Symp on Foun

dations of Computer Science pp

A Lemp el S Even I Cederbaum An algorithm for planarity testing of a graph

Theory of Graphs International Symp osium Gordon and Breach New York

Y Maon B Schieb er U Vishkin Paral lel ear decomposition search EDS and st

numbering in graphs TCS pp

G Miller V Ramachandran A new graph triconnectivity algorithm and its paral leliza

tion ACM Symp on Theory of Computing pp

T Nishizeki N Chiba Planar graphs theory and algorithms North Holland

V Ramachandran J Reif An optimal paral lel algorithm for graph planarity Pro c

th Ann IEEE Symp on Found of Comp Sci pp

B Schieb er U Vishkin On nding lowest common ancestors simplication and par

al lelization Pro c AWOC LNCS SpringerVerlag pp

R Tamassia and J S Vitter Parallel transitive closure and p oint lo cation in planar

structures SIAM J Comput pp

RE Tarjan UVishkin An ecient paral lel biconnectivity algorithm SIAM J Com

puting pp

C Thomassen Planarity and duality of nite and innite planar graphs J Combina

torial Theory Series B pp

WT Tutte Convex Representations of Graphs Pro c London Math So c vol

WT Tutte How to Draw a Graph Pro c London Math So c vol no pp

Algorithms for Emb edding Graphs Into a page Bo ok

Miki Shimabara Miyauchi

A book is a twopart ob ject that consists of a spine which is a line and pages each of

which is a halfplane b ounded by the spine A book embedding of a graph orders the no des

of the graph linearly along the the spine of a b o ok and arranges each edge on pages so

that the edges do not intersect Bo ok emb eddings have applications in several areas of

theoretical computer science including VLSI design and complexity theory In the

Diogenes metho d for designing faulttolerant VLSI pro cessor arrays for example a b o ok

emb edding is used to implement the live pro cessors of a multilayer chip the pro cessors

extended through all the layers The singlerow routing problem is also considered to

b e b o ok emb edding problem having two pages and questions such as how to minimize the

numb er of lines crossing the spine of the b o ok and how to minimize the numb er of tracks

of a graph have b een investigated

The b o okemb edding problem has b een studied for several kinds of graphs When b o ok

emb eddings are restricted so that each edge is emb edded on one page for example M Yan

nakakis showed that any planar graph can b e emb edded in a b o ok of four pages and

Chung Leighton and Rosenb erg showed that the complete graph K is emb eddable in

n pages When each edge can b e emb edded in more than one page Atneosen Babai

and Bernhart have each shown that every graph is emb eddable in a page b o ok

Atneosens pro of however is nonconstructive By using Leightons lower b ound on the

crossing numb er of a complete graph we show the following

Theorem Babai and Bernharts algorithm that embeds K into a page book takes

n time

NTT Basic Research Lab oratories MidoriCho MusashinoSh i Tokyo Japan mikinttnttjp

We also present a new emb edding algorithm for page b o ok emb eddings of graphs

Theorem There is an algorithm that embeds any graph into page book and that runs

in time O mn for a graph of size m and order n

Sketch of the proof Let D b e a half disk with center c on the xy plane and let C b e

the b oundary of D Arrange the n no des V fv g on C counterclo ckwise

for each edge e v v n s i E do

s i

if s i or i then draw v v as a straight line segment on D

s i

else let p b e the middle p oint of the straight line segment v v

s s s

and p b e the p oint at which the straight line segment p v

crosses the auxiliary line c v Join the pair of p oints fv p g by a straight

i s

line segment on D and join the pair of p oints fp v g by a half circle

in a plane p erp endicul ar to D

Let c b e the p oint at which the auxiliary line c v crosses the b oundary of a small neigh

i i

b orho o d of c The path v c c v v c v is considered as the spine L of the

b o ok D is considered two pages and the third page is taken as the sharply b ent surfaces

p erp endicular to D The for statement rep eats m jE j times and each inner for lo op

rep eats at most n jV j times

Theorem Any algorithm embedding K into a page book takes n time

Theorem Let K be a complete graph with n s s I s nodes Then there

is an embedding of K into a page book with n n n edgecrossings

over the spine of the book

References

G A Atneosen On the Emb eddability of Compacta in NBo oks PH D Thesis

Michigan State Univ

F Bernhart The Bo ok Thickness of a Graph J Combi Theory Ser B pp

F R K Chung F Thomson Leighton and A L Rosenb erg Emb edding Graphs in

Bo oks A Layout Problem with Applications to VLSI Design SIAM J Alg Disc Math

Vol No

Z Galil R Kannan E Szemeredi On nontrivial sep erations for kpage graphs and

simulations by nondeterministic onetap e Turning machines th ACM Symp on

Theory of Computing pp

F T Leighton New Lower Bound Techniques for VLSI Math Systems Theory

T T Trarng M MarkekSadowska and E S Kuh An Ecient SingleRow Routing

Algorithm IEEE Tran Computeraided Design Vol CAD No

A T White Graphs Groups and Surfaces pp NorthHol land

M Yannakakis Emb edding Planar Graphs in Four Pages J Computer and System

Sciences pp

Dominance Drawings of Bipartite Graphs

y y z y

Hossam ElGindy Michael Houle Bill Lenhart Mirka Miller David

Rappap ort and Sue Whitesides

A partial order P of a nite set X is a transitive and nonreexive binary relation on

X A partial order can b e represented by a transitive digraph G on the elements of X

The dimension dP of a partial order P is the minimum numb er of linear orders whose

intersection is P There is a direct interpretation of dP as it p ertains to its asso ciated

graph We can use vectors x x x x to represent each vertex x of G so that x

k i

y i k with strict inequality in at least one co ordinate if and only if y is reachable

from x in G Such an assignment of co ordinates to vertices s called a dominance drawing

b ecause all edges in the graph and transitive closure are geometrically characterized by

the dominance relation We use the notation dG to denote the dominance drawing

dimension of the graph G

In it is proved that deciding whether dP is NPcomplete A characterization

in of partial orders of dimension two or less leads to a p olynomial time recognition

algorithm Thus let G denote the complement of an undirected version of the transitive

G is transitively orientable that is digraph representing P Then dG if and only if

the edges of G can b e oriented to obtain a transitive digraph Transitive orientability can

b e tested in O K time where denotes the maximum vertex degree and K the numb er

of edges in the graph

We present an algorithm that obtains a two dimensional dominance drawing of a transitive

bipartite graph whenever such a drawing exists The running time of the algorithm is

prop ortional to the size of the input and is thus optimal This compares favourably with

the complexity of the algorithm in Let G S T E b e a transitive bipartite graph

with S s s s a set of sources and T t t t a set of sinks We use

jS j jT j

N s ft s t E g to denote the neighbourhood of s Let N fN s s S g and

i j

let M fN s N w s w S N w N sg then I S T N M Let I S T

denote the collection of all p ermutations of T such that memb ers of each subset I

I S T are contiguous in Our algorithm to obtain a two dimensional dominance drawing

of the graph G is based on the following characterization of two dimensional transitive

bipartite graphs

Theorem The dimension of G S T E is less than or equal to two if and only if

I S T is not empty

Part of the work was carried out when the authors were participants of the Workshop on Layout and Optimal

Path Problems at Hawks Nest Australia sp onsored by the Department of Computer Science The University of

Newcastle New South Wales Australia

Department of Computer Science The University of Newcastle New South Wales AUSTRALIA

Department of Computer Science Willia ms College Williamstown USA

Department of Computing and Information Science Queens University Kingston Ontario KL N

CANADA

Scho ol of Computer Science McGill University Montreal Queb ec CANADA

Consider a set X and a set of subsets of X Bo oth and Leuker present the so called

PQtree algorithms that can b e used to determine the family of p ermutations of X so that

every subset is contiguous in the family of p ermutations The resulting computational

complexity is linear in the size of the input Thus it app ears that an exp edient solution to

our problem is to compute the set M and subsequently the set I S T and apply the PQ

tree algorithm However the size of M may b e O jS j This problem is not insurmountable

as we can restrict our attention to a linear sized subset of M Consider the case where we

have a maximal sequence N s N s N s then we only need to consider the

set N s N s However the task of computing this reduced subset of M approaches

the conceptual diculty of presenting a new approach without using PQtrees We present

a new algorithm and skirt the problem of computing a reduced subset of M The data

structure we use to represent the family of p ermutations I S T is inuenced by the

PQtree but it is sp ecially tailored for this problem and is much simpler

We dene a boxlist of a set T B T as the empty list or a linked list consisting of one

or more b oxes where each b ox contains a subset of T and the b oxes form an exact cover

of T that is the union of the b oxes in the b oxlist is T and each element of T app ears

in exactly one b ox If we x intrab ox ordering of elements then a rear to front or front

to rear traversal of B T corresp onds to a p ermutation of T A p ermutation of T is

consistent with B T if the elements within b oxes can b e ordered so that a traversal of

B T is equal to Let F B T b e used to denote the set of all p ermutations that are

consistent with B T Given a graph G S T E our algorithm b egins with a b oxlist

representing all p ermutations of T that is the b oxlist consists of exactly one b ox that

contains T itself If the graph has a two dimensional dominance drawing then the algorithm

exits with a nonempty b oxlist such that F B T I S T otherwise the algorithm

exits with B T an empty list

The principle op eration p erformed on a b oxlist is to add constraints to B T that are

asso ciated with a neighb ourho o d of a source N s The constraints are substrings or

intervals within the p ermutations of I S T Thus B T is constrained so that the

interval asso ciated with N s will b e contiguous in all p ermutations of F B T We show

that the adding of constraints can b e scheduled so that it is easy to check whether an

interval is contiguous within B T and that N s N w is contiguous for all w such that

N w N s We maintain an overall linear complexity by avoiding explicit sorting

Our algorithm development is summarized in the following theorem

Theorem Given a connected transitive bipartite graph G S T E our algorithm

returns F B T I S T The algorithm can be implemented to run in O jS j jT j

jE j time and space and this is within a constant multiple of optimal

References

G Di Battista WP Liu and I Rival Bipartite Graphs Upward Drawings and

Planarity Information Pro cessing Letters vol pp

K Bo oth and G Leuker Testing for the consecutive ones property interval graphs

and graph planarity using PQtree algorithms J Comput and Sys Sci

B Dushnik and E W Miller Partial ly ordered sets Amer J Math pp

M Golumbic Algorithmic Graph Theory and Perfect Graphs Academic Press

A Pnueli A Lemp el and S Even Transitive orientation of graphs and identication

of permutation graphs Canad J Math pp

R Tamassia Drawing algorithms for planar stgraphs Australasian Journal of Com

binatorics vol

M Yannakakis The complexity of the partial order dimension problem SIAM J

Alg and Disc Meth Vol No pp

Computing the Overlay of Regular Planar Sub divisions in

Linear Time

Ulrich Finke and Klaus Hinrichs

A planar subdivision is the emb edding of a planar graph in the plane and therefore

determines a partitioning of the plane It is an op en problem whether the overlay of two

simply connected sub divisions can b e pro cessed in linear time and space A rst step to

solve this problem is the overlay algorithm for convex sub divisions which has the desired

complexity Generalizations of this result are of theoretical and practical imp ortance

We prop ose an top ological sweepline algorithm which solves the problem of overlaying

regular sub divisions in linear time and space A regular sub division is simply connected

the emb edding of each edge curve must b e a function in x and each vertex v of the planar

graph has at least one incoming edge from the left and one outgoing edge to the right

The sweepline in our algorithm divides the overlay sub division into two parts The left

part is the correctly pro cessed overlay and the right part contains the unpro cessed elements

of the sub divisions All edges b etween the left and right part intersect the top ological

sweepline The basic idea of our algorithm is to handle only those edges that represent the

seams b etween the sub divisions on the sweepline By p erforming lo cal sweepline transactions

the algorithm sews up the sub divisions to generate the overlay result The lo cality of the

transactions makes it p ossible to parallelize the algorithm easily

Algorithms for computing the overlay of planar sub division are of great practical imp ortance

in geographic information systems and computational geometry

References

B Chazelle Computational Geometry for the Gourmet Old Fare and New Dishes

ICALP

A U Frank Overlay Pro cessing in Spatial Information Systems Pro c th Int Sym

p osium on ComputerAssisted Cartography AUTOCARTO

FB Informatik Westfalische WilhelmsUniversi tat Einsteinstr D Munster Germany

nkehinrichsmathuni muens terde

L J Guibas R Seidel Computing Convolutions by Recipro cal Search nd ACM

Symp osium on Computational Geometry

F P Preparata M I Shamos Computational Geometry An Intro duction Springer

Verlag New York

C D Tomlin Geographic Information Systems and Cartographic Mo deling Prentice

Hall Englewo o d Clis NJ

Generation of Random Planar Maps

Alain Denise

Metho ds of random generation are useful to ols to study some prop erties of combinatorial

structures As regards graphs such metho ds are ecient to verify or formulate conjectures

esp ecially when the exhaustive generation of all the graphs which are to b e studied would

b e unreasonable These metho ds also allow to evaluate the p erformances of algorithms on

such structures Moreover to increase ones own knowledge on some class of graphs it

is useful to b e able to generate and to display these congurations Thus pro cedures of

random generation are included in softwares of manipulation of graphs which are used for

research and teaching of graph theory and discrete mathematics The uniform generation

of random graphs has b een well studied for a few years and ecient algorithms exist for

some particular classes See for exemple works of Tinhofer Dixon and Wilf Wormald

Jerrum and Sinclair

The matter of our work is the uniform generation of random ro oted planar maps with n

edges A planar map is the pro jection of a planar connected graph on a plane surface A map

is ro oted if a vertex and an edge adjacent to it are distinguished By using the enco ding

of planar maps due to Cori and Vauquelin we reduce the problem to a problem of

generation of words of a language close to the language of parenthesis systems Then we

use a rejection algorithm inspired by the metho ds of Barcucci Pinzani and Sprugnoli

in order to generate these words We prove that the average complexity is O n and we

conjecture that it is O n n

References

E Barcucci R Pinzani and R Sprugnoli Generation aleatoire des animaux diriges In

J G Penaud J Lab elle editor Actes de lAtelier FrancoQuebecois de Combinatoire

publi LaCIM Universite du Queb ec a Montreal

E Barcucci R Pinzani and R Sprugnoli Generation aleatoire de chemins sous

eme

diagonaux In C Reutenauer P Leroux editor Actes du Col loque Series Formel les

et Combinatoire Algebrique publi LaCIM Universite du Queb ec a Montreal

O Baudon Cabrigraphes un cahier de brouil lon interactif pour la theorie des graphes

PhD thesis Universite Joseph Fourier Grenoble

LaBRI Universite Bordeaux I Talence Cedex France deniselabriub ord eaux fr

R Cori and B Vauquelin Planar maps are well lab eled trees Can J Math

H S Wilf and J D Dixon The random selection of unlab eled graphs J Algorithms

M Jerrum and A Sinclair Fast uniform generation of regular graphs TCS

G Tinhofer On the use of almost sure graph prop erties Lecture Notes in Computer

Science

N C Wormald Generating random regular graphs J Algorithms

Symmetric Drawings of Graphs

Joseph Manning

Intro duction

Perhaps one of the most imp ortant criteria for pro ducing visuallyinformative drawings of

abstract graphs is the display of axial andor rotational symmetry collectively known as

geometric symmetry Its imp ortance stems from the fact that given a symmetric drawing

an understanding of the entire graph can b e built up from that of a smaller subgraph

replicated a numb er of times This pap er reviews several results on the denition detection

and display of geometric symmetry

Geometric symmetry in graphs may b e dened by using either a geometric or an algebraic

formulation Geometrically a graph is said to p ossess a particular symmetry if there exists

some drawing of the graph which displays that symmetry Note however that geometric

symmetry is an inherent prop erty of the abstract graph itself rather than of any individual

drawing Algebraically a geometric symmetry may b e dened as an automorphism of the

graph which satises certain conditions Both denitions are equivalent While all of the

results b elow were obtained from the geometric denition it app ears that the algebraic

denition may hold the greatest promise in attempting to expand these results to broader

classes of graphs

The fundamental problem of determining if a general abstract graph p ossesses any geometric

symmetry along with several variations are all NP complete Accordingly the current

research has fo cused on symmetry in planar graphs since these constitute an imp ortant

sub class of general graphs and frequently admit ecient solutions to otherwise intractable

problems

Vassar College USA manningcsvassaredu

Algorithms

Optimal lineartime algorithms outlined b elow have b een develop ed for detecting and

displaying b oth axial and rotational symmetries in the following classes of planar graphs

Trees A tree free tree is a connected acyclic graph All symmetries of a

tree must keep its center xed A center of a tree is any vertex whose maximum

distance from any leaf is minimized every tree has either one center or two adjacent

centers the latter case may b e reduced to the former by intro ducing a new vertex

on the edge joining the two centers Removing the center from a tree divides the

remainder of the tree into a numb er of subtrees and any symmetry of the overall

tree must p ermute these among themselves mapping subtrees to isomorphic subtrees

Using a variation of the lineartime treeisomorphism test Mp these subtrees

are partitioned into isomorphism classes from which the geometric symmetries are

subsequently determined A radial drawing of the tree which displays these symme

tries is then constructed Since it is imp ossible in general to display all of its axial

symmetries in a single drawing of a given tree the algorithm instead determines the

maximum numb er of simultaneouslydisplayable axial symmetries and constructs the

corresp onding most symmetric drawing

Outerplanar Graphs An outerplanar graph is one which can b e drawn in the

plane with no edge crossings and with all vertices on the outer face Every biconnected

outerplanar graph has a unique Hamilton cycle which may b e found in linear time

By traversing its Hamilton cycle the graph is transformed into a string in such a

way that geometric symmetries of the graph corresp ond to certain symmetries of

the string These in turn are found using an ecient patternmatching algorithm

from which the symmetries of the graph are then recovered By contrast with

the situation for trees all geometric symmetries of a biconnected outerplanar graph

may b e displayed in a single drawing which the algorithm then constructs For non

biconnected outerplanar graphs geometric symmetries are enumerated by using a

combination of the ab ove algorithm applied to the biconnected comp onents with the

algorithm for trees applied to the blo ckcutvertex tree while a similar combination of

drawing techniques is used to construct symmetric drawings

Plane Embeddings of Planar Graphs A planar graph is one which can b e drawn

in the plane with no edge crossings and a plane emb edding merely lists the order

of edges emanating from each vertex without sp ecifying either the co ordinates of

the vertices or the shap es of the edges A planar graph with such an emb edding is

rst transformed into a numb er of concentric biconnected outerplanar levels and

the geometric symmetries of these levels are then found using the previous algorithm

and intersected to give the symmetries of the original graph This algorithm has

particular relevance to drawing triconnected planar graphs whose plane emb eddings

are unique up to the choice of outer face

Many of the algorithms have b een implemented and test runs have shown that their optimal

theoretical time complexities do indeed translate into fast practical algorithms Running on

a midrange workstation a graph with up to one hundred vertices can b e pro cessed almost

instantaneously

Extensions

Perhaps the most imp ortant challenge lies in extending these results to the entire class of

planar graphs An easier extension might b e to seriesparallel graphs which form another

prop er sub class of planar graphs

The incidence of geometric symmetry in graphs app ears to decrease as the size of the graph

increases For example while of vertex trees have at least one axial symmetry only

of vertex trees do It app ears useful to relax the requirement of strict symmetry

and instead investigate using nearsymmetry as a drawing criterion An even more far

reaching generalization would b e to explore the construction of straightedge drawings in

which the numb er of distinct edge lengths is minimized

References

A Aho J Hop croft J Ullman The Design and Analysis of Computer Algorithms

AddisonWesley

D Knuth J Morris V Pratt Fast Pattern Matching in Strings SIAM Journal on

Computing Jun

J Manning M Atallah Fast Detection and Display of Symmetry in Trees Congressus

Numerantium Nov Also available as Technical Rep ort CSDTR

Department of Computer Sciences Purdue University Dec

J Manning M Atallah Fast Detection and Display of Symmetry in Outerplanar

Graphs Discrete Applied Mathematics Aug Also available as Tech

nical Rep ort CSDTR Department of Computer Sciences Purdue University Mar

J Manning Computational Complexity of Geometric Symmetry Detection in Graphs

Lecture Notes in Computer Science SpringerVerlag Jun Also available as

Technical Rep ort CSC Department of Computer Science University of Missouri

Rolla Jan

J Manning Geometric Symmetry in Graphs PhD Thesis Department of Computer

Sciences Purdue University Dec

Recognizing Symmetric Graphs

Tomaz Pisanski

Usually we can deal with graphs if they are small and we are able to grasp their pictorial

representation If a graph is stored and the information of its construction is lost the problem

is how to recognize the graph how to nd its optimal or nearoptimal construction For a

human the drawing of a graph may represent a way of recognizing the graph

IMFM Department of Theoretical Computer Science University of Ljubljana Jadranska Ljubljana

Slovenia tomazpisanskiuni ljsi

We implemented a series of algorithms for automatic drawing of graphs The rst metho d

uses the idea of spring embedding by P Eades Ead that comes in several variants

Graduate student Danica Dolnicar wrote a Pascal program that compares the algorithm of

Kamada and Kawai KK to the metho d of Fruchterman and Reingold FR

In the second one we exp erimented with eigenvectors We noticed that the nd rd and

th eigenvector can b e used as the three co ordinates for the vertices in the dimensional

space Later we obtained some theoretical results that will b e presented in a joint pap er

with John ShaweTaylor

For vertex transitive graphs we tried to use a combination in order to pro duce go o d results

for large graphs First we calculate the automorphisms of graph We use B McKays Nauty

to do the job Then we select an automorphism that has the smallest numb er of orbits

We contract the vertices in each orbit of and thus obtain the factor graph Then the factor

graph is drawn using an automatic drawing algorithm Finally the orbits are blown out so

that the vertices are put on the cycles in the order sp ecied by the p ermutation For

instance the Coxeter graph on vertices that is otherwise hard to recognize is drawn in

the familiar Y shap e If all the orbits are of the same size say k another drawing approach

may b e taken The vertices of the k copies of the factor graph are placed on a circle in order

to display rotational symmetry Then the edges of the original graph are drawn as straight

lines The idea of displaying symmetry in graph drawing is certainly not new The reader

is referred to and for further information ab out the research on this topic

It should b e noted that computationintensive algorithms are b eing programmed by students

and researchers at IMFM in computer languages such as Pascal and C however they are

all bundled in a Mathematica Wat package called Vega The whole system will b ecome

available for nonprot use in

References

M J Atallah and J Manning Fast Detection and Display of Symmetry in Emb edded

Planar Graphs Manuscript Purdue Univ

G Di Battista P Eades R Tamassia I G Tollis Algorithms for Drawing Graphs

an Annotated Bibliography Preprint June

P Eades A Heuristic for Graph Drawing Congressus Numerantium vol pp

TMJ Fruchterman and EM Reingold Graph Drawing by Forcedirected Place

ment SoftwarePractice and Exp erience vol pp

T Kamada and S Kawai An Algorithm for Drawing General Undirected Graphs

Information Pro cessing Letters vol pp

J Manning and M J Atallah Fast Detection and Display of Symmetry in Trees

Congressus Numerantium vol pp

J Manning and M J Atallah Fast Detection and Display of Symmetry in Outerplanar

Graphs Technical Rep ort CSDTR Dept of Computer Science Purdue Univ

West Lafayette IN

S Wolfram Mathematica AddisonWesley Redwo o d City CA

Algorithmic and Declarative Approaches to Aesthetic

Layout

Peter Eades and Tao Lin

Aesthetics for graph layout can b e divided into three categories

Global criteria such as minimizing the numb er of edge crossings or maximizing sym

metry

Correctness criteria such as placing the employer ab ove the employees in an organi

zation tree drawing

Preferred criteria which express the preferences of a sp ecic user at a sp ecic time

These criteria may include placing a particular no de in the centre of the page or using

a particular asp ect ration

We can divide implementations of layout functions into two general categories those with

an algorithmic approach and those with a declarative approach

The algorithmic approach is well do cumented in the survey This approach concen

trates on achieving global criteria Typically the approach ignores the semantic or syntactic

meaning of a sp ecic diagram and only uses graph theoretic structure

In a layout algorithm the aesthetics are hard co ded into the implementation of the function

It is not easy to change the requirements for a layout algorithm at run time Such layout

algorithms are not exible cannot cop e with preferred criteria and normally can only cop e

with a few xed correctness criteria

A declarative layout function handles the layout according to a exible set of require

ments sp ecied by endusers or interface designers even at run time A system which uses

declarative layout creation function has two comp onents an editor through which user can

sp ecify aesthetics and a mechanism for creating the layout The aesthetics may b e presented

as constraints rules or parameters for a cost function There are several mechanisms used

in the declarative approach such as constraint solvers genetic systems rule based systems

and simulated annealing

The expressive p ower of constraints and rules ensure that a wide variety of requirements

can b e sp ecied thus the declarative approach maximizes exibili ty

However the declarative approach has signicant problems It is dicult to cho ose the rules

or constraints it is dicult to foresee the eects of a constraint or rule even in a mo derately

sized system Implementations of declarative functions are very slow and sometimes due

to their heuristic nature do not achieve their aim even there is a layout which satises

relevant requirements In particular the declarative approach has some diculty handling

global aesthetics

We conclude that neither algorithmic nor declarative approaches are suitable for sole adop

tion in graphic user interfaces However by integrating the approaches one can use the

advantages of one to comp ensate the disadvantages of the other

If a layout algorithm is built on top of a declarative system the integrated system seems very

slow Declarative techniques are used successfully on top of algorithms in TBB

Department of Computer Science University of Newcastle Newcastle Australia eadescsnewcastleeduau

frankcsnewcastleeduau

however in these cases the declarative techniques are tightly b ound to the algorithms We

b elieve that an eective approach is to lo osely tie some declarative functions on top of a

layout algorithm Briey this may b e achieved by exploiting the nondeterminism in layout

algorithms

The integrated system may b e used as follows

The interface modeler creates a generic to olb ox of layout algorithms covering a wide

range of global aesthetics The algorithms leave p oints of nondeterminism to b e ex

ploited by constraints

The interface designer cho oses a sp ecic set of algorithms for a sp ecic application The

algorithms are customized by sp ecifying constraints so that they satisfy the correctness

criteria of the application

The end user sp ecies further constraints to achieve preferred criteria

This approach satises the same set of the global criteria as the underlying layout algo

rithm However the integrated approach is suciently exible to supp ort a broad range of

correctness and preferred criteria

We give examples of this approach using tree drawings

References

KarlFriedrich Bohringer and Frances Newb ery Paulisch Using constraints to achieve

stability in automatic graph layout algorithms In Proceedings of ACMSIGCHI pages

P Eades and R Tamassia Algorithms for drawing graphs an annotated bibliography

Technical rep ort Department of Computer Science Brown University Accepted

to the Networks

EB Messinger Automatic Layout of Large Directed Graphs PhD dissertation De

partment of Computer Science University of Washington Published as Technical

Rep ort No

Rob erto Tamassia Giusepp e Di Battista and Carlo Batini Automatic graph drawing

and readability of diagrams IEEE Transactions on Systems Man and Cybernetics

JanuaryFebruary

A Visual Approach to Graph Drawing

y y y

Isab el F Cruz Rob erto Tamassia and Pascal Van Hentenryck

This abstract describ es research in progress on a new technique for the visual sp ecication

of constraints in graph drawing systems

Work in graph drawing has traditionally fo cused on algorithmic approaches where the

layout of the graph is generated according to a presp ecied set of general rules or aesthetic

criteria such as planarity or area minimization that are emb o died in an algorithm Perhaps

the most sophisticated graph drawing system based on the algorithmic approach is the one

develop ed by Di Battista et al BBL which maintains a large database of graph drawing

algorithms and is able to select the one b est suited to the needs of the user

The algorithmic approach is computationally ecient however it do es not naturally sup

p ort constraints ie requirements that the user may want to imp ose on the drawing of a

specic graph eg clustering or aligning a given set of vertices Previous work by Tamassia

et al TBB has shown the imp ortance of satisfying constraints in graph drawing sys

tems and has demonstrated that a limited constraint satisfaction capability can b e added

to an existing drawing algorithm Recently several attempts have b een made at developing

languages for the sp ecication of constraints and at devising techniques for graph drawing

based on the resolution of systems of constraints Kam Mar HM

Current constraintbased systems have three ma jor drawbacks

The sp ecication of constraints is made through a detailed enumeration of facts from

a xed set of predicates expressed in Prolog Kam or with a settheoretic notation

Mar

Natural requirements such as planarity need complicated constraints to b e expressed

General constraintsolving systems are computationally inecient HM

The ab ove discussion indicates the need for a language to sp ecify constraints that reconciles

expressiveness with eciency

We b elieve that visual languages could provide a natural and userfriendly way to express

the layout of a graph For this purp ose we plan to design a variation of doodle a visual

language for the sp ecication of the display of facts in an ob jectoriented database Cru

Cru We envision the following goals which dierentiate our work from Mar and

Kam

Visual sp ecication of layout constraints the user should not have to typ e a long list

of textual sp ecications

Extensibility the user should not b e limited to a presp ecied set of primitives

Flexibility the user should not have to give precise geometric sp ecications such as

exact co ordinates or precise geometric relations

Research supp orted in part by the National Science Foundation under grant CCR by the US Army

Research Oce under grant DAALG and by the Oce of Naval Research and the Advanced Research

Pro jects Agency under contract NJ ARPA order

Department of Computer Science Brown University Providence RI fifcrtpvhgcsbrownedu

In addition to constraints the visual language should also b e able to express aesthetic

criteria asso ciated with optimization problems eg crossing or area minimization and to

identify general drawing standards eg layered drawing or upward drawing Recent work

by Eades and Lin EL has similar ob jectives but is not based on a visual sp ecication

For eciency reasons we envision using our visual language within a graph drawing sys

tem similar to Diagram Server BGST A crucial comp onent of this system is a com

piler that translates the visual sp ecications into a drawing algorithm synthesised from a

database of drawing algorithms The algorithms database will contain b oth p olynomial and

exp onentialtime algorithms The main purp ose of the drawing compiler is to deduce from

the sp ecications a combination of algorithms that solves the layout problem as eciently

as p ossible The work in BBL is particularly interesting in this context

References

BBL P Bertolazzi G Di Battista and G Liotta Parametric Graph Drawing Tech

nical Rep ort Consiglio Nazionale delle Ricerche Rome Italy July

BGST G Di Battista A Giammarco G Santucci and R Tamassia The Architecture

of Diagram Server In Proc of IEEE Workshop on Visual Languages

Cru Isab el F Cruz DOODLE A Visual Language for Ob jectOriented Databases

In ACMSIGMOD Intl Conf on Management of Data pages

Cru Isab el F Cruz Using a Visual Constraint Language for Data Display Sp ecica

tion In Paris Kanellakis JeanLouis Lassez and Vijay Saraswat editors First

Workshop on Principles and Practice of Constraint Programming

EL Peter Eades and Tao Lin Algorithmic and Declarative Approaches to Aesthetic

Layout In Proc of Graph Drawing

HM Richard Helm and Kim Marriott Declarative Sp ecication of Visual Languages

In Proc IEEE Workshop on Visual Languages

Kam Tomihisa Kamada Visualizing Abstract Objects and Relations A Constraint

Based Approach World Scientic Singap ore

Mar Jo e Marks AFormal Sp ecication for Network Diagrams That Facilitates Au

tomated Design Journal of Visual Languages and Computing

TBB R Tamassia G Di Battista and C Batini Automatic Graph Drawing and

Readability of Diagrams IEEE Transactions on Systems Man and Cybernetics

SMC

Layout of Trees with Attribute Graph Grammars

Gaby Zinmeister

Graph grammars are a formalism for syntactically describing classes of graphs Their pro

duction rules are used for graph rewriting to implement graph manipulations There are a

variety of application areas ranging from software development environments and compiler

construction over pattern recognition and development of biological cell layers to sp ecica

tion of concurrent systems

The central idea of our approach to layout graphs is viewing layout algorithms as attribute

evaluators of attribute graph grammars thus a layout algorithm is an attribute scheme plus

an attribute evaluator The main advantage is that dierent layouts can b e sp ecied simply

with dierent attribute schemes

We have mo delled three widely used tree layout algorithms with attribute graph

grammars AGG A graph grammar GG for arbitrary trees is describ ed The three dier

ent layout algorithms are sp ecied by dierent attribute schemes of the same grammar We

illustrate our approach with the Mo en algorithm All three algorithms will b e presented

in the full pap er

Attribute Graph Grammars AGG

GGs are a generalization of string grammars which are well known from formal language

theory Kreowski and Rozenb erg give an excellent survey of graph grammars For

our purp oses we use so called context free no de lab el controlled GGs CFNLC The

pro ductions of such a GG consist of one nonterminal no de as the lefthand side of the rule

a graph over nonterminal and terminal no des as the righthand side and an emb edding

sp ecication A derivation step consists of cho osing one o ccurrence of the lefthand side

nonterminal A in the host graph H which is the analogue of a sentential form removing

that instance adding the righthand side graph R and connecting the remainder of H to R

following the emb edding sp ecication The emb edding sp ecication describ es which no des

of the neighb ourho o d of the instance of A in H are to b e connected with which no des of

R Thus edges may b e deleted or added or their orientation may b e inverted The language

pro duced by a GG is an innite set of graphs

The AGG approach used in this pap er is a generalization of Knuths attribute string

grammars Synthesized or inherited attributes are asso ciated to no des and attribute eval

uation rules to the GG pro ductions Attribute evaluators may b e constructed analogously

see For sp ecial GGs ecient p olynomial time parsers can b e constructed and

attribute evalutor generators have b een implemented

The GG for Arbitrary Trees

The GG for trees is dened as follows

Let T fF S g ftg fxg P S F with the pro ductions as in g

N T E

WilhelmSchi ckardInsti tut Sand D Tubingen zinssmeiinformati kuni tuebi ng ende

p p p

S F S

p p p

out out p out out

p p p

fF S tg x x fF S tg x x

p p p

in in p in in p in in p

fF S tg x x fg fS tg x x fF S tg x x

p p p

S F S

Figure Pro ductions P of T

S no des are the generic no des with which we can add an arbitrary numb er of children to

a treeno de whereas F no des represent a fully derived subtree

The upp er nonterminal no des are the left hand sides the lower graphs with terminal and

nonterminal no des and solid edges are the right hand sides of the pro ductions The dotted

in in

lines indicate the replacement step The mark fF S tg x x means that xlab elled

edges from F S and tno des to the left hand side no de are replaced by xlab elled edges

out

from the same sources to the right hand side no de sp ecied by the dotted line x denotes

the reverse orientation of edges The empty set fg at a dotted line denotes that the target

no de is not connected to the host graph

The terminal alphab et can obviously b e extended but is reduced to one no de typ e for

simplicity There is only one edge lab el so we can think ab out the terminal tree as one with

unlab elled edges

The Mo en Algorithm as AGG

The main idea in attributing is to push information ab out subtree contours from b ottom

to top in the derivation tree synthesized attributes joining subtrees in the S no des and

adjusting the ro ot over the subtrees in the F no des The relative p osition of a no de to

its sibling is calculated in this pass to o Afterwards in a second pass the missing relative

p ositions of the rst child of each subtree are pushed downwards in the derivation tree

inherited attributes Absolute co ordinates may b e assigned to no des in this second pass

as well but are omitted here for simplicity Figure lists the attributes and gure shows

the complete attribution The functions written in italic are taken from the original Mo en

algorithm except for dierent use of reference parameters and global values The attributes

are evaluable with a two pass left to right evaluation on the derivation tree

Conclusion

The only other approach we know to drawing graphs using graph grammars is published

by Brandenburg He augments graph grammar pro ductions by placement rules in form

of leftof aboveof constraints b etween r hsno des a b ounding b ox around the r hs and

some connection p oints on that b ounding b ox This is well suited for graphs where replacing

a nonterminal no de results pushing the rest of the graph in x and y dimension to get

space for the r hsno des at the relative old place of the l hsno de Such graphs are for

example syntax diagrams or series parallel graphs But with the tree graph grammar ab ove

constraints in the sense Brandenburg will not lead to a reasonable layouts b ecause it is not

Attributes Symb ol Typ e Explanation

ro otdims syn F S integers width height and b order of the ro ot no de

contour syn F p olyline contour around the tree derived from F

contoursubtrees syn S p olyline contour around the subtrees derived from S

lastsubtreeheight syn S integer height incl b order of the subtreero ot derived last from S

sum syn S integer sum of heights of the subtrees

dims syn t integers width height b order of the terminal no de

oset inh S F t p osition relative x yco ordinates

heightsubtrees inh S integer height of subtrees derived from S and its siblings

Figure Attributes for the Mo en Algorithm

p ossible to dene leftof relations b etween siblings The reason is that sibling no des are

not related by edges and not pro duced within the same pro duction so no relation can b e

stated

As further work we plan to develop a CFNLC GG for directed acyclic graphs DAGs

and attribute it with some of the current layout algorithms for DAGs Even though the

literature describ es generation of p olynomial time parsers for sub classes of GGs

and generation of attribute evalutors for AGGs no running implementation is available

For that reason we are implementing a parser and an attribute evaluator for T GG For

interactive environments where graphs are manipulated incremental layout of graphs is

necessary This could b e done by applying incremental attribute evaluation to the attribute

graph grammar approach

References

F J Brandenburg Layout Graph Grammars the Placement Approach In H Ehrig

HJ Kreowski and G Rozenb erg editors GraphGrammars and Their Application

to Computer Science Int Workshop Bremen Germany LNCS pages

Springer

D Janssens and G Rozenb erg Graph Grammars with No deLab el Controlled Rewriting

And Emb edding In H Ehrig M Nagl and G Rozenb erg editors GraphGrammars

and Their Application to Computer Science Int Workshop Haus Ohrbeck Ger

many LNCS pages Springer

M Kaul Syntaxanalyse von Graphen b ei PrazedenzGraphGrammatiken Technical

Rep ort MIP Univ Passau dissertation

HJ Kreowski and G Rozenb erg On Structured Graph Grammars I Information

Sciences

HJ Kreowski and G Rozenb erg On Structured Graph Grammars I I Information

Sciences

S Mo en Drawing Dynamic Trees IEEE Software July

E M Reingold and J S Tilford Tidier Drawings of Trees IEEE Trans Softw Eng

SE Mar

A Schutte Spezikation und Generierung von Ubersetzern fur GraphSprachen durch

attributierte GraphGrammatiken Reihe Informatik EXpress Edition Berlin

dissertation

p F ro otdims S ro otdims

F contour IF S contoursubtrees empty THEN layout leafS ro otdims

ELSE attach parentS contoursubtrees S ro otdimsS sum

S oset F oset

S heightsubtrees S sum

p S contoursubtrees IF S contoursubtrees empty THEN F contour

2 0 1

ELSE mergeS contoursubtreesF contour

S lastsubtreeheight F ro otdimsheight F ro otdimsb order

S ro otdims S ro otdims

0 1

S sum IF S contoursubtrees empty THEN S lastsubtreeheight

0 1 0

ELSE S lastsubtreeheight mergeS contoursubtreesF contour S sum

0 1 1

F osetx IF S contoursubtrees empty THEN

parent distance S ro otdimswidth S ro otdimsb order const

1 1

ELSE

F osety IF S contoursubtrees empty THEN

S ro otdimsheight S heightsubtrees S ro otdimsb order

1 1 1

ELSE mergeS contoursubtreesF contour S lastsubtreeheight

1 1

S oset S oset

1 0

S heightsubtrees S heightsubtrees

1 0

p S contoursubtrees empty

S lastsubtreeheight

S sum

S ro otdims tdims

toset S oset

Figure Attribute Scheme of T for the Mo en Layout Algorithm

TreeWidget of the Athena Widget Set XR distribution of MIT

The Display Browsing and Filtering of Graphtrees

y y

Sandra P Foubister and Colin Runciman

Context

We are writing an interpreter for a little lazy functional language Implementation is by

graph reduction The user is allowed to view and explore the program graph at every

reduction step or at less frequent intervals on request The aim is to gain insight into the

pro cess of lazy graph reduction where the order of reduction is not always intuitive There

are two main ob jectives to explore the extent of sharing and to b e able to identify areas

of ineciency The p otential size and complexity of the graphs p ose problems for display

This pap er presents and discusses novel solutions to these problems

Complexity

One way of simplifying the display is to avoid any crossing of arcs There is no guarantee

that a program graph will b e planar indeed the features of a lazy language sharing

recursion and knot tying in general make planarity unlikely Rather than trying to

display every arc in the graph the solution b eing investigated is to use a spanning tree

This is enhanced with display leaves to represent arcs that would otherwise not b e shown

Display leaves are lab eled with a reference to the vertex to which they represent an arc

The problem of program graph display is thus limited to that of tree display The sp ecial

kind of tree b eing displayed is referred to as a graphtree

Size

The problem of size comp ounded by the addition of display leaves may b e resolved in

several ways the scale of the display may b e reduced or only part of the graph may b e

shown at a time In addition to these a solution prop osed here is that the size of the

underlying graph b e reduced by grouping vertices together in clusters so that the new

graph has fewer vertices

Graphtrees

The implementation of the programming environment is itself in a lazy functional language

namely Haskell In such a language one can dene a displayable graphtree typ e DGT

that is convenient for subsequent display and browsing of the structure It is parametrised

on index value and reference typ es Indices uniquely identify vertices values are vertex

lab els not necessarily unique and the reference typ e is the typ e of the display reference

typically an integer

Supp orted by the Science and Engineering Research Council of Great Britain

Department of Computer Science University of York Heslington York YO DD

fsandracolingmin steryorkacuk

data DGT i v r DGT Xpos Vertex i v r DGT i v r NoDGT

The Xpos is a provisional p osition on the x axis of the display that may b e scaled to an

actual x co ordinate

The Vertex is either a reference to a DGT not instantiated at this p oint or a vertex value

with its asso ciated identifying index and p ossibly a display reference

data Vertex i v r Ref DGT i v r Val v i Maybe r

The list of DGTs within a DGT construction comprises a predecessor as well as the successors

of the current DGT we have a threaded structure that exploits the laziness of the dening

language in its construction

NoDGT is needed to represent the predecessor of the ro ot of the directed graph Each DGT

has directly available sucient information to redisplay the graphtree with itself at the

ro ot

Display

The requirements of the display are that it should b e compact but also revealing of the

structure Various styles of presentation were considered includin g the tipover and inclu

sion conventions describ ed in and the p ossibility of showing the tree as a free tree see

despite the existence of a ro ot For other purp oses these may b e suitable but in our

system the display of a graphtree reects the conventional display of applicative expressions

as trees interior vertices corresp ond to applications with function and argument graphs

as successors Shared reference arcs may p oint back up the tree

A mo dication of Vauchers algorithm is used to calculate Xpos entries as the DGT is

b eing created The nal display is a spanning tree of the graph with an extra no de for

each arc that is not part of the tree The choice of spanning tree is determined by the

order in which vertices are visited during the display routine At present the order is that

resulting from our variant of Vauchers algorithm but the resulting structure may not b e

ideal for ltering and browsing and may not have the most satisfying app earance However

determining an optimal spanning tree for display purp oses may b e infeasibly complex in

our interactive setting

Browsing

In addition to the main display a minigraph scaled to t exactly onto a small window is

used as a map for browsing as advo cated by Beard and Walker The graphtree has the

shap e it would have if lab els were present for concordance with the main display but no

lab els are shown

The main display is in a larger window but on a xed scale so the graphtree may have

to b e pruned Arcs to vertices o the display are truncated to form stubs Clicking on the

display of a vertex or the end of a stub or a display reference brings the appropriate vertex

to the ro ot of the display Clicking in the minigraph window p ermits the user to jump to

another section of graphtree

Filtering

In order to reduce the numb er of vertices in the graph to b e displayed without violating

the meaning of the original graph the notion of a homosemantic graph is intro duced The

idea is that a cluster of vertices with their interconnecting arcs b ecomes one vertex in a

graph of clusters This vertex inherits all the arcs from the vertices it incorp orates that

connect with the rest of the graph The value of the new vertex integrates the values of its

constituent vertices The full structure of the original graph is not retained in the display

but the conditions under which clusters may assimilate others are dened in such a way

that the graph has the same meaning The implementation of such a ltering scheme raises

various interesting questions ab out the denition of suitable lters and the ordering of

compaction of the graph

The system outlined ab ove oers an eective way of observing even large and complex

graphs Our current goal is to provide users of our application with a exible mechanism

for dening lters to achieve such views of the program graph as they nd necessary

References

L Allison Circular programs and selfreferential structures Software Practice and

Experience

David V Beard and John Q Walker I I Navigational techniques to improve the display

of large twodimensional spaces Behaviour and Information Technology

Peter Eades Drawing free trees Technical rep ort International institute for advanced

study of so cial information science Fujitsu lab oratories Ltd Japan

Peter Eades Tao Lin and Xuemin Lin Two tree drawing conventions International

Journal of Computational Geometry and Applications

Jo e Fasel Paul Hudak Simon Peyton Jones and Phil Wadler Sp ecial issue on the

functional programming language Haskell ACM SIGPLAN Notices

Simon Peyton Jones and David Lester Implementing Functional Languages Prentice

Hall

Kenneth J Sup owit and Edward M Reingold The complexity of drawing trees nicely

Acta Informatica

Jean G Vaucher Prettyprinting of trees Software Practice and Experience

A Layout Algorithm for Undirected Graphs

Daniel Tunkelang

We prop ose an algorithm for generating straightline two dimensional layouts of undirected

graphs Our algorithm uses a combination of heuristics to obtain layouts which are near

optimal with resp ect to an aesthetic cost function The heuristics improve on existing

approaches by fo cusing on three asp ects of the graph layout problem computation of the

aesthetic cost of a layout order of no de placement and lo cal optimization techniques

An implementation of our algorithm in C on an IBM RS workstation lays out most

graphs of up to no des in under seconds many in less than seconds and consistently

generates layouts which with resp ect to the three aesthetic criteria describ ed b elow are

b etter than those pro duced by the forcedirected algorithm of Fruchterman and Reingold

FR and the simulated annealing algorithm of Davidson and Harel DH

Di Battista et al DBETT discusses three aesthetic criteria for drawing graphs edge

lengths should b e uniform nonadjacent no des should b e far away from each other and

the numb er of edge crossings should b e minimal The rst two criteria are characteristic of

the spring embedder mo del prop osed by Eades Ead and further develop ed by Kamada

and Kawai KK and Fruchterman and Reingold FR Our algorithms aesthetic cost

function quanties a weighted evaluation of all three criteria

Our algorithm has three stages In the rst stage it determines the order in which no des will

b e placed by constructing a minimal height breadthrst spanning tree of the graph This

ordering enumerates the no des of the graph from the center outwards In the second stage

the algorithm places the no des one at a time by sampling p ositions near the already placed

neighb ors of a no de After placing each no de the algorithm lo cally optimizes the layout

near that no de The optimization pro cess propagates itself through neighb oring no des that

is whenever the lo cal optimization pro cedure succeeds in improving the placement of a

no de it calls itself recursively on all of the already placed neighb ors of that no de After

this pro cess stabilizes at a lo cal optimum the algorithm pro ceeds iterating through the

list of no des The third stage netunes the layout by again p erforming lo cal optimization

at every no de

Our algorithms sp eed is largely the result of the way it computes the aesthetic cost function

First the computation is incremental so that a small change in the layout requires mini

mal recomputation Second the algorithm approximates the cost by ignoring interactions

b etween far away nonadjacent no des as in the gridvariant principle of Fruchterman and

Reingold FR Third the algorithm uses the uniform grid technique of Akman et al

AFKN to compute edge crossings Although these metho ds are not conceptually origi

nal none of the published graph layout algorithms apply them in combination to computing

the aesthetic cost function which is the inner lo op of computation

Unlike most of the published layout algorithms for undirected graphs which initially place

all no des randomly our algorithm places no des one at a time in a deterministic order The

inspiration for our metho d is a no deordering strategy prop osed by Watanab e Wat Our

algorithm places the no des in an order that reects their centrality in the graph thereby

minimizing the constraints on the no des which will eventually o ccupy the denser regions of

Carnegie Mellon University For a complete pap er please send email to DanielTunkelan gcscmuedu This

work is based on my Masters Thesis which was sup ervised by Charles Leiserson at MIT and Mark Wegman at

the IBM T J Watson Research Center

the layout This strategy exemplies a general principle a deterministic strategy based on

knowledge of the problem is b etter than a random one based on ignorance

The other innovation in our algorithm is its lo cal optimization metho d Whenever the algo

rithm improves the layout by moving a no de it propagates the lo cal optimization pro cess to

that no des neighb ors That is whenever the improvement pro cedure nds a way to reduce

the cost asso ciated with a particular no de it eects that improvement and then calls itself

recursively on all of that no des neighb ors This approach provides several b enets First

p erfect initial placement is not so imp ortant b ecause the immediate attempt at lo cal opti

mization netunes the initial guess Secondly propagating optimization through neighb ors

tends to nd the regions of the layout that need improvement and concentrate on them

Thirdly this metho d of lo cal optimization ensures that the algorithm gets what it pays for

the time it sp ends in lo cal optimization is b ounded in terms of the numb er of improvements

the optimization generates This metho d of lo cal optimization together with a metho d for

rapid initial placement makes the algorithm fast and eective

The pro of of our algorithms merit is in its p erformance Fruchterman and Reingold as

well as Davidson and Harel graciously allowed implementations of their algorithms to

b e used for comparison with an implementation of ours The test suite for comparison

consisted of examples from their pap ers as well as textb o ok examples of graphs and

randomly generated graphs of up to no des The prop osed algorithms running time is

ab out the same as that of Fruchterman and Reingold and is much faster than the simulated

annealing algorithm of Davidson and Harel All three algorithms were aiming for the same

aesthetic criteria uniformity of edge lengths distribution of no des and edge crossings Our

algorithm consistently pro duced the b est layout with resp ect to these measures esp ecially

the numb er of edge crossings A drawback of our algorithm is that it do es not explicitly

consider the angles b etween adjacent edges since measuring them would have required

oating p oint computation As a result some of these angles are almost illegibly small In

general however our algorithm is very eective at pro ducing lowcrossing aesthetically

pleasing drawings of graphs of up to no des and average degree of up to four or ve

Beyond this the graphs b ecome to o dense for the prop osed algorithm and for those cited

as well

In summary our algorithm improves on existing work by optimizing the inner lo op of com

putation intelligently cho osing an order for no de placement and using a lo cal optimization

strategy that exploits lo cal structure within a layout We continue to explore the many

op en problems in graph layout

References

AFKN V Akman W R Franklin M Kankanhalli and Narayanaswami Geometric

computing and uniform grid technique ComputerAided Design

Septemb er

DBETT G Di Battista P Eades R Tamassia and I G Tollis Algorithms for drawing

graphs an annotated bibliography Technical rep ort Brown University June

DH R Davidson and D Harel Drawing graphs nicely using simulated annealing

Technical rep ort Department of Applied Mathematics and Computer Science

Weizmann Institute of Science April Revised version

Ead P Eades A heuristic for graph drawing Congressus Numerantium

FR T Fruchterman and E Reingold Graph drawing by forcedirected placement

Software Practice and Experience Novemb er

KK T Kamada and S Kawai An algorithm for drawing general undirected graphs

Information Processing Letters April

Wat H Watanab e Heuristic graph displayer for gbase International Journal of

ManMachine Studies

Drawing Ranked Digraphs with Recursive Clusters

Stephen C North

Abstract not Available

Graph Drawing Algorithms for the Design and Analysis of

Telecommunication Networks

Ioannis G Tollis and Chunliang Xia

The problem of drawing a graph in the plane has received increasing attention recently due

to the large numb er of applications Examples include VLSI layout algorithm animation

visual languages and CASE to ols Vertices are usually represented by p oints and edges

by simple op en curves In this pap er we study techniques for visualizing telecommunication

networks The visualization of telecommunication networks is very useful in aiding the

design pro cess of minimum cost networks and the management of network op erations

We present linear time algorithms for drawing telecommunication networks with optimal

area so that imp ortant prop erties are displayed

The design and analysis of cost eective survivable telecommunication networks is a very

imp ortant problem Most problems that aim towards minimizing the total

cost of a network are NPhard For that matter computer to ols to aid the design and

analysis of telecommunication networks are in great demand A central problem of such

to ols is how to draw a network on the computer screen such that imp ortant asp ects of

the network can b e easily captured and an improved solution can b e obtained by a user

interactively

The problem is dened as follows Let G V E b e a telecommunication network with

a set of no des V representing the sites of switches and a set of links E representing

The authors are with the Department of Computer Science The University of Texas at Dallas Richardson

TX tollisutdal las edu xiautdall asedu

the electrical wires or optical b er links b etween no des The trac requirements b etween

the no des are dened by an n n matrix T where T i j corresp onds to the amount of

trac b etween no des i and j We need to design a network which a satises the trac

requirements b can survive failures and c the cost of the network is minimum A

network is survivable if it can survive the failure of a link e ie the removal of link e

do es not disconnect the network and the trac that originally travels through e can b e

accomo dated on another path The multiring architecture is considered as a costeective

survivable network architecture due to its simplicity improved survivability and bandwidth

sharing A ring cover of G is a set of rings cycles of G such that the rings are

connected and every no de in V is included in at least one ring Apparently a network with

a ring cover is survivable since the switches automatically send the required trac around

the ring if a link failure o ccurs

Since the no des of the network corresp ond to sites they have geographic co ordinates Hence

the network can b e drawn naturally with little eort However the imp ortant prop erties

of the network that designers are interested in such as rings are not displayed In this

pap er we describ e several algorithms for drawing telecommunication networks in order to

to aid the design of costecient networks Given a ring cover of a network our algorithms

display it in such a way that rings are easily identiable and p ossible problems can b e easily

sp otted by network designers

Ideally we want to draw all rings as cycles but this is not always p ossible if rings are

not allowed to intersect For instance if three rings share a common no de then the cycles

will intersect In cases like this we will use a geometric shap e with a slight deviation from

cycle called almostcycle to represent rings An almostcycle is such a geometric shap e

that except for very few no des almost all no des of a ring are placed on the b oundary of a

cycle Even if we allow rings to cross not all ring covers admit such a representation In this

pap er we present a necessary condition for ring covers that admit such a representation

As is the case in most graph drawing algorithms the exsitence of unnecessary crossings is

viewed as harmful to the readability of the drawings Thus minimizing such crossings is

central to our approach Also we assume the exsitence of a resolution rule that is in the

nal drawing any two no des of the network must b e kept far enough so that the human

eye can tell them apart This implies that the drawing cannot b e arbitrarily scaled down If

we honor such a resolution rule and represent rings as cycles there is a trivial lower b ound

of N on the area required for the drawing where N is the numb er of no des in the

network

In order to capture the complexity of interaction among rings a new graph G is intro duced

Given a network G and its ring cover C a contact node is a no de of G that is contained in at

least two dierent rings of C Let V b e the collection of all contact no des G C V E

where E fr v j r C v V and v is a no de of ring r g Graph G is called the ring

contact node graph According to the denition of a ring cover G is a connected graph

When G is a tree three dierent drawing algorithms are intro duced outside drawing

inside drawing and mixed drawing All of the algorithms create zero unnecessary crossings

and take linear time

In the rest we present our main results

It seems natural to put two rings side by side when they share a contact no de since all

rings will b e drawn on the outer space we call this style of drawing outside drawing

Theorem Algorithm outside drawing results in a drawing which takes O N area and

each ring is represented by an almostcycle

Instead of placing two rings side by side when they share a contact no de we place one

inside another We call this style of drawing inside drawing

Theorem Algorithm inside drawing results in a drawing which takes O N area and

each ring is represented by a cycle

There are some cases where an inside drawing outp erforms an outside drawing There are

also cases where an outside drawing outp erforms an inside drawing Hence we combine the

strength of the two to obtain a mixed drawing

We assign a weight to each no de v in G If v is a contact no de v has weight if v is a

ring v has a weight equal to the numb er of no des in the ring Let N b e the length of the

second longest path in G

Theorem Algorithm mixed drawing results in a drawing which takes O N N area

and each ring is represented by a cycle

References

G Di BattistaP Eades R Tamassia and I G Tollis Algorithms for Automatic Graph

Drawing An Annotated Bibliography Dept of Comp Science Brown Univ Technical

Rep ort Available via anonymous ftp from wilmacsbrownedu

les pubgdbibliotexZ and pubgdbibliopsZ

G Di Battista E Pietrosanti R Tamassia and IG Tollis Automatic Layout of

PERT Diagrams with XPERT Pro c IEEE Workshop on Visual Languages VL

GR Dattatreya JP Fonseka K Kiasaleh IH Sudb orough IG Tollis and S

Venkatesan Advanced Network Top ologies for Network Survivability Technical Re

p ort UTD January

MR Garey and DS Johnson Computers and Intractability A Guide to the Theory

of NPCompleteness Freeman

WD Grover BD Venables JH Sandham and AF Milne Performance Studies

of a SlefHealing Network Proto col in Telecom Canada Long Haul Networks IEEE

GLOBECOM pp

G Kar B Madden and R S Gilb ert Heuristic layout Algorithms for Network Man

agement Presentation Services IEEE Network Novemb er

H Sakauchi Y Nishimura and S Hasegawa A SelfHealing Network with an eco

nomical SpareChannel Assignment IEEE GLOBECOM pp

J C Shah Restoration Network Planning To ol Pro c th Annual Fib er Optic Engi

neers Conf April

TH Wu DJ Collar and RH Cardwell Survivable Network Architectures for Broad

band Fib er Optic Networks Mo del and Performance Comparisons IEEE Journal of

Lightwave Technology Vol No Novemb er pp

TsongHo Wu and R C Lau A Class of SelfHealing Ring Architectures for SONET

Network Applications IEEE Trans on Communication Vol No Nov

A View to Graph Drawing Algorithms through GraphEd

Michael Himsolt

We compare a collection of graph drawing algorithms implemented in our Graph system

We rep ort on our exp erience from running these algorithms on a large numb er of exam

ples b oth from the literature and by our own and present our evaluation of the practical

relevance of the algorithms and layout criteria

The representation of complex structures as graphs is widespread Graph drawing has gained

increasing imp ortance in many areas of Computer Science but has proved to b e a dicult

Ed Ed

task Our Graph system is an approach to supp ort solutions to this problem Graph

has b een used by practitioners for database design Petri nets and electrical circuits One

of its ma jor applications is the implementation and evaluation of graph layout algorithms

With its capabilities to create and edit graphs Graph provides an eective environment

to create and test large sets of examples Since all drawing algorithms are built into one

to ol it is easy to compare the eect of dierent algorithms on the same graph

There is also a sp ecial mo dule layout suite that runs all applicable layout algorithms on

one graph It also writes statistical data such as the size of the graph the space used or the

numb er of b ends and crossings We have created a large database of graphs and statistics

with that mo dule

We regard testing many examples as an adequate and probably the b est way to get precise

data on the practical relevance of layout criteria and graph drawing algorithms Currently

the following algorithms are implemented

Spring emb edder based on algorithms by FruchtermannReingold and Kamada

Tree drawing Walker

Dag drawing SugiyamaTagawaTo da

Planar drawing on a grid Wo o ds

Planar drawing on a grid with b ends minimization Tamassia

Planar straightline drawing with convex faces ChibaOnoguchiNishizeki

Planar straightline drawing on a grid de FraysseixPachPollack and

ChrobakPayne

Drawing Petri nets from term descriptions Seisenb erger

We have tested these algorithms on a large numb er of examples b oth from literature and

by our own We have tested arbitrary graphs as well as graphs with sp ecial structure eg

grids Our exp eriences can b e comprised as follows

Spring emb edders pro duce go o d layouts for most graphs They stress the display of

isomorphic and symmetric substructures A ma jor drawback is the high runtime

From the practical p oint of view tree drawing seems to b e solved since the algorithm

repro duces the tree structure in the same way as the user would do it by hand

The algorithm of SugiyamaTagawaTo da provides a go o d base for drawing dags but

the layouts are not as go o d as trees

Universitat Passau Passau himsoltfmiuni passau de

Planar graphs are generally dicult to draw b ecause planarity alone do es not ex

plain the intrinsic structure of the graph Moreover some planar graph drawing algo

rithms dep end on the actual planar emb edding which causes further problems

Tamassias algorithm gives our b est layouts although it has the highest running time

in this class The drawings lo ok pretty

Wo o ds algorithm gives suitable results for small graphs but the numb er of b ends is

much higher as with Tamassias algorithm

The algorithms of ChibaOnoguchiNishizeki and de FraysseixPachPollack are of

limited practical use They tend to cluster no des and often destroy pleasing pictures

The Petri net algorithm takes agents as input which are term descriptions of the

nets It pro duces very go o d layouts This comes from the fact that the agents provide

detailed information on the structure of the nets The information is used to draw the

graph as a designer would do

The go o d results stimulate our work on a general framework of graph grammar based

layout algorithms

From the exp eriments our actual ranking of layout criteria is

Distribute the no des in a uniform fashion

Display the intrinsic structure of the graph

Display symmetric and isomorph substructures of the graph

Use few edge crossings to draw the graph none if the graph is planar

Use few b ends to draw the graph

Place no des and b ends on a grid

Straight line edges may or may not b e a go o d criterium dep ending on the other criteria

used in a particular algorithm It often imp oses restrictions on the layout like in the al

gorithms of ChibaOnoguchiNishizeki and de FraysseixPachPollack We have found out

that allowing a few b ends as in Tamassias algorithm is usually a go o d choice

Graph is available with anonymous ftp from forwissunipassaude

publocalgraphed

References

G Di Battista P Eades R Tamassia and IG Tollis Algorithms for Drawing Graphs

An Annotated Bibliography contains references on most of the algorithms mentioned

ab ove

M Himsolt Konzeption und Implementierung von Grapheneditoren Dissertation

Universitat Passau to b e published

K Seisenb erger Termgraph ein System zur zeichnerischen Darstellung von struk

turierten Agenten und Petrinetzen Diplomarb eit Universitaet Passau

An Automated Graph Drawing System Using Graph

Decomp osition

y z z

C L McCreary C L Combs D H Gill and J V Warren

Intro duction

This pap er presents a graph layout technique CG based on the hierarchial decomp osi

tion of graphs One of the ma jor dierences b etween the graphs drawn by CG and other

systems is that the vertices in CGs graphs are spaced in a balanced way b oth vertically

and horizontally Many other systems partition vertices into levels and all vertices of the

same level are placed on the same horizontal axis No des tend to bunch toward the top

of the graph is these systems By using our graph decomp osition technique CG is able to

determine balanced vertical spacing as well as balanced horizontal spacing Edge crossings

are reduced by a very ecient variant of the Barycentric metho d that exploits the subgraph

hierarchy

Graph Decomp osition

Clanbased graph decomp osition is a parse of a directed acyclic graph DAG into a

hierarchy of subgraphs called clans A subset X of DAG G is a clan i for all x y X and

all z G X a z is an ancestor of x i z is an ancestor of y and b z is a descendant of

x i z is a descendant of y

A simple clan C with more than three vertices is classied as one of three typ es It is

i primitive if the only clans in C are the trivial clans ii in dependent if every subgraph

of C is a clan or iii linear if for every pair of vertices x and y in C x is an ancestor or

descendant of y Indep endent clans are sets of isolated vertices which can b e visualized as

horizontal neighb ors Linear clans are sequences of one or more vertices v v v v

i i j j

where for i k v is an ancestor of v and can b e seen as a vertical string Any graph

i k

can b e constructed from these simple clan as well as decomp osed into a parse tree with

clan comp onents Primitive clans do not fall into the clearcut categories of vertices that

should b e laid out horizontally or laid out vertically One pro cedure for further reduction

of primitive clans is to form an indep endent clan of the source vertices of the primitive and

decomp ose the remainder of the primitive The indep endent clan is linearly connected to

the rest of the clan The parse tree of any completely decomp osed graph is a bipartite tree

where the internal vertices represent clans that are classied as either linear or indep endent

The parse tree of the graph can b e given a geometric interpretation A b ounding rectangle

with known width and height can b e asso ciated with each clan The parse tree hierarchy

shows the emb edding of the b ounding rectangles

A simple twodimensional algebra denes the bounding rectangles Singleton DAG vertices

or equivalently parse tree leaves have unit square b ounding rectangles Linear clans require

Dept of Computer Science and Engineering Auburn University Auburn AL mccrearyengauburnedu

Equifax Incorp orated Atlanta Georgia

The MITRE Corp oration McLean Virginia

an area whose length is the sum of the lengths of the comp onent clans and whose width

is the maximum width of the comp onent clans Indep endent clans require an area whose

width is the sum of the widths of the comp onent clans and whose length is the maximum of

the lengths of the comp onent clans To achieve an aesthetically pleasing layout the vertices

are centered within the b ounding rectangles Since clans are dened as groups of vertices

with identical connections to the rest of the graph clans can easily b e contracted to a single

vertex Any vertex not in the clan that was connected to a clan vertex will b e connected

to the contracted vertex By allowing segments of the graph to b e contacted the user

can simplify graphs for viewing by contracting those parts which are not relevant to her

investigation Contracted vertices can b e expanded to show the original clan conguration

The Barycentric Metho d Adapted to Clans

The Barycentric metho d a heuristic for reducing the numb er of edge crossings in two

consecutive levels of a graph is mo died by considering adjacent clans instead of adjacent

vertices The pro cess pro ceeds by rearranging adjacent parse tree children of the largest

unpro cessed linear clan Because groups of vertices clans rather than individual vertices

are sub ject to rearrangement at each step the metho d is much more ecient than the

standard Barycentric metho d

By insp ecting the structure of a graph through graph decomp osition an aesthetically pleas

ing and natural layout of the graph vertices can b e constructed By adapting existing edge

routing techniques CG is able to draw arcs that have few unnecessary edge

crossings and that are smo oth and straight

References

E R Gansner S C North and K P Vo Dag a program that draws directed acyclic

graphs Software Practice and Experience Novemb er

D H Gill T J Smith T E Ohasch C L McCreary and I V Warren R E K Stire

walt Sparialtemp oral analysis of program dep endence graphs for usefull parallelisim

Journal of Paral lel and Distributed Computing to app ear

D Jablownowski and V A Guana Jr Gmb A to ol for manipulating and anomating

graph data structures Software Practice and Experience March

L A Rowe et al A browser for directed graphs Software Practice and Experience

January

K Sugiyama S Tagawa and M To da Metho ds for visual understanding of hi

erarchichal system structures IEEE Transactions on Syst Man and Cybernetics

February

J N Wareld Crossing theory and hierarchy mapping IEEE Transacrions on Syst

Man and Cybernetics July

Maximum Planar Subgraphs and Nice Emb eddings

Practical Layout To ols

Michael Junger and Petra Mutzel

In automatic graph drawing a given graph has to b e layed out in the plane p ossibly accord

ing to a numb er of top ological and aesthetic constraints Nice drawings for sparse nonplanar

graphs can b e achieved by determining a maximum planar subgraph and augmenting an

emb edding of this graph Finding maximum planar subgraphs is NPhard and therefore

this technique app eared not to b e practical

We attack the problem with techniques of p olyhedral combinatorics The p olytop e P LS G

of G is dened as the convex hull over all incidence vectors of planar subgraphs of G and

called the planar subgraph p olytop e The problem of nding a planar subgraph P of G

with weight w P as large as p ossible can b e written as the linear program maxfw x j x

P LS Gg since the vertices of the p olytop e P LS G are exactly the incidence vectors of

the planar subgraphs of G In order to apply linear programming techniques to solve this

linear program one has to represent P LS G as the solution of an inequality system Due

to the NPhardness of our problem we cannot exp ect to b e able to nd a full description

of P LS G by linear inequalities But even a partial description of the facial structure of

P LS G by linear inequalities is useful for the design of a branch and cutalgorithm

b ecause such a description denes a relaxation of the original problem Such relaxations

can b e solved within a branch and b ound framework via cutting plane techniques and linear

programming in order to pro duce tight b ounds For a partial description by inequalities we

only have to concentrate on prop er faces of maximal dimension of P LS G socalled facet

dening inequalities One of the main results of our investigation of the facial structure of

the planar subgraph p olytop e is the fact that all the sub divisions of K or K turned out

to b e facetdening for P LS G

We have designed a branch and cut algorithm using facetdening inequalities for P LS G

as cutting planes In a cutting plane algorithm a sequence of relaxations is solved by linear

programming After the solution x of some relaxation is found we must b e able to check

whether x is the incidence vector of a planar subgraph in which case we have solved the

problem or whether any of the known facetdening inequalities are violated by x If no such

inequalities can b e found we cannot tighten the relaxation and have to resort to branching

otherwise we tighten the relaxation by all facetdening inequalities violated by x which we

can nd Then the new relaxation is solved etc The pro cess of nding violated inequalities

if p ossible is called separation or cutting plane generation Although the vectors x

coming up as solutions of LPrelaxations in the ab ove outlined pro cess have fractional

comp onents in general they are often useful to obtain information on how a highvalued

planar subgraph might lo ok like We exploit this idea with a greedy typ e heuristic with

resp ect to the solution values of the edges So in addition to the upp er b ounds w x on the

value of a maximum planar subgraph we also obtain a lower b ound w x from the planar

subgraph incidence vector x derived heuristically from x The lower b ound heuristic as well

as the cutting plane generation are based on a planarity testing algorithm of Hop croft and

Tarjan

In our computational exp eriments we solved several problems from the literature to opti

mality Among them there is a graph given by Tamassia Di Battista and Batini in a pap er

Institut fur Informatik Universitat zu Koln Pohligstrae Koln Germany mjuengerinformatikuni

ko elnde mutzelinformatikuni ko el nde

ab out automatic graph drawing TBB In order to get the maximum planar subgraph

of the graph the algorithm removed four of the edges The computation to ok seconds

on a SUN SPARCstation mo del For the graph given by Kant in K on no des

and edges the algorithm found an optimum solution with edges in seconds

In order to explore the limits of our branch and cut algorithm we tested it on a series of

randomly generated graphs We could observe that the easiest problem instances are those

on sparse graphs which are almost planar and dense graphs We could observe that our

co de is able to solve all problem instances with up to edges to optimality Even though

we cannot solve all instances of bigger sizes to optimality our approach allows us to give

quality guarantees which state that our solutions are less than p b elow the optimum

where p is given when the computation stops after a certain amount of time The quality

guarantee turns out to b e typically less than for random problems with up to edges

References

HT Hop croft J and RE Tarjan Ecient planarity testing J ACM

K Kant G An On Maximal Planarization Algorithm based on PQtrees

Utrecht University

TBB Tamassia R G Di Battista and C BatiniAutomatic graph drawing and

readability of diagrams IEEE Transactions on Systems Man and Cybernetics

Heuristics for Planarization by Vertex Splitting

Peter Eades and Xavier Mendonca

Intuitively a vertex v may b e split by making two copies v and v and attaching the

edges incident with v to either v or v The op eration is illustrated in Figure

| | | |

| |

e e e e

1 2 1 2

e e

3 3

| | |

v v

1 2

S S

e e

5 5

S S

e e

4 4

S S

| |

S S

| S | S

Figure The spliting op eration

This simple op eration is intro duced to change the graph a little to make it amenable to

layout

Eective layout algorithms imp ose restrictions on the input graph structure The two main

sources of this problem are

layout algorithm design The layout algorithm is limited to sp ecial classes of graphs For

instance there is a wealth of layout algorithms for planar graphs however these

algorithms are useless for nonplanar graphs

task diculty The layout task has high complexity and we must imp ose restrictions on

the input to b e able to handle the task within reasonable computational resources For

instance nding a planar drawing of a graph in which all edges have length one is

in general NPhard However when the input is restricted to trees there are trivial

layout algorithms for this aesthetic

The problem of transforming the input graph to conform with the restrictions is an im

p ortant concern These transformations should change the graph as little as p ossible so

the graph do es not lose it identity For instance to planarize a graph we may delete

a small numb er of edges or add a small numb er of dummy vertices at crossings Many

optimisation problems of minimising the numb er of such transformations are NPcomplete

Men but eective heuristics are available for some

We are concerned with the splitting transformation describ ed ab ove Manual layout tech

niques sometimes involve making a copy of a no de in order to simplify layout see for

example Lim We aim to investigate the automation of layout techniques using the

splitting op eration

We are particularly interested in four basic aesthetic criteria planarity edge length sym

metry and straight line drawing

We present a heuristic for planarization by splitting which we call SPLITPLANARIZE The

SPLITPLANARIZE heuristic is based on Lemp el Even and Cederbaums planarity test

ing algorithm LEC its implementation using PQtrees BL and the PLANARIZE

algorithm of Jayakumar Thulasiraman and Swamy JTS

We also present two other algorithms which use vertex splitting for dierent ob jectives The

rst algorithm TENSIONSPLIT is a heuristic which consists of a mo dication of a spring

system A tension is calculated for each eligible vertex and after each lo cal minimization

step a vertex with high tension is split The ob jective of the TENSIONSPLIT algorithm is

to pro duce a layout of a graph G transformed from G by splitting in which the Euclidean

distance b etween the pairs of vertices u and v in G is equal to the length of the edges uv

Finally an optimization criterion to p erform splitting using simulated annealing is pre

sented This is a very interesting approach since several dierent criteria can b e applied to

pro duce dierent emb eddings

References

BL K S Bo oth and G S Lueker Testing for the consecutive ones prop erty inter

val graphs and graph planarity using pq tree algorithms J Comp Syst Sci

Dec

JTS R Jayakumar K Thulasiraman and M N S Swamy on algorithms for graph

planarization IEEE Trans ComputerAided Design Mar

LEC A Lemp el S Even and I Cerderbaum An algorithm for planarity testing of

graphs In Theory of Graphs Int Symp pages Rome Italy July

P Rosenstiel Ed Gordon Breach

Lim MIM Holdings Limited Organization Charts MIM Holdings Limited Queens

land Australia Internal Memorandum

Men C F X De Mendonca ALayout System for Information System Diagrams PhD

dissertation The University of Queensland Department of Computer Science

JuneAugust

Planar Graph Emb edding with a Sp ecied Set of

FaceIndep endent Vertices

Takao Ozawa

Intro duction

It is known that there are many ways in general to emb ed a biconnected planar graph in

the plane In this pap er we intro duce a new graph emb edding problem as dened b elow

and give a very ecient solution algorithm to it Let G b e a biconnected planar graph with

vertex set V

Problem FIVSEMB Given a subset U of V nd if p ossible an emb edding of G in the

plane such that no two vertices of U app ear on a face b oundaryeach vertex in U is covered

by a distinct face

In graph theory two vertices are said mutually indep endent if they are not the end vertices

of an edge Extending the concept of indep endence in relation with edges to that in relation

with faces we say that two vertices not app earing on a face b oundary are faceindep endent

If we want some edges in addition to vertices b eing faceindep endent we only have to place

a new vertex on each of the edges so that each edge is converted to a series connection of

two edges incident to the new vertex and then include the newlyadded vertices in U In

integrated circuit layout it may happ en that some elements should not b e placed closely to

avoid mutual interference We can say that the graph emb edding of FIVSEMB takes such

a constraint into consideration in a simplied way

Solution Algorithm

Our solution algorithm to problem FIVSEMB is based on the vertex addition algorithm for

planarity testing and is implemented using PQtrees The vertices of G are lab eled

with the stnumbers and thus we have Vf ng where n is the numb er of vertices

in V Let G b e the subgraph of G induced by the vertex set V f k g Roughly

k k

sp eaking the vertex addition algorithm successively emb eds G for k n in the

plane

Now a vertex pair fxygfxyg is not equal to fng is called a separation pair if the

removal of the pair results in a disconnected graph Let C b e a connected comp onent of the

resultant graph containing vertices whose stnumb ers are b etween x and y The subgraph

Department of Applied Mathematics and Informatics Ryukoku University Seta Ohtsu Siga Japan

ozawarinsryukokuacjp

of G which is obtained by adding x and y to C is called an fxygsplit comp onent There

may b e two or more split comp onents for the pair fxyg and there may b e an edge or

edges connecting vertices x and y Let Sxy b e the set which consists of all fxygsplit

comp onents and edges connecting x and y if any Dierent emb edding can b e obtained

by the following op erations permutation changing the emb edding order of the split

comp onents and edges b elonging to Sxy and reection reecting biconnected split

comp onents in Sxy

For PQtrees op erations on no des corresp onding to the ab ove op erations on Sxy are

dened A subgraph formed by Sxy with a xed emb edding order of split comp onents

and edges in it is called a comp osite split comp onent

In our solution algorithm we try to nd an emb edding of G which satises the condition

of problem FIVSEMB Diculty arises when there are two or more ways of emb edding

satisfying the condition and the entire emb edding of G can not b e nalized We present two

ma jor subalgorithms coping with this diculty The rst one of them nds by applying

the op erations of p ermutation and reection the most desirable emb edding of G while

leaving some part of the emb edding undecided In order to implement this subalgorithm

lab els which indicate the existence of vertices in U on the b oundaries of split or comp osite

split comp onents are attached to the no des of the PQtree representing G and the sub

algorithm nds the most desirable lab els of no des while bubbling up the PQtree The

second ma jor subalgorithm decides using an auxiliary bipartite graph the emb edding

of split or comp osite split comp onents which are previously formed and contained in the

relevant split comp onent but whose emb edding has not b een nalized We also present a

subalgorithm for nding the vertices contained in a split comp onent while dealing with

PQtrees This subalgorithm is necessary for carrying out the second ma jor subalgorithm

mentioned ab ove

The time complexity and the space complexity of our solution algorithm are b oth On

References

A Lemp el S Even and I Cederb oum An algorithm for planarity testing of graphs

in Theory of Graphs International Symp osium Rome July pp Gorden

and Breach N Y

K S Bo oth and G S Lueker Testing for consecutive ones prop erty interval graphs

and graph planarity using PQtree algorithms J Computer and System Sciences vol

Implementation of the Planarity Testing Algorithm by

Demoucron Malgrange and Pertuiset

Bjorn Sigurd Benestad Johansen

Previous Planarity Algorithms Hop croft Tarjan and Bo oth Leuker have

presented quite complicated linear planarity algorithms A much simpler algorithm has

b een presented by Demoucron et al The algorithms by Hop croft Tarjan and Demou

cron et al will b e referred to henceforth as HT and DMP resp ectively

Although DMP is basically simple it is abstract and no data structures or details are

given The goal of this authors implementation of DMP DMPHT has b een to provide a

fastyetsimple planarity algorithm mo del

A Straightforward Implementation of DMP As describ ed in Bondy Murty a

straightforward implementation of DMP requires subroutines for

Finding a cycle in the inputgraph G

Determining the fragments in BondyMurty called bridges of G in G and their vertices

G where G is a plane subgraph of G of attachment to

i i

G Determining the b oundary of each region in

Determining for each fragment F the regions of G where F is drawable

G b etween two of F s Vertices of Attachment Finding a path p in some fragment F of

Previous Implementations of DMP Rubin has given an O n implementation

of DMP implementing to a large extent the Subroutines ab ove Rubin claims that his

implementation compares favorably to HT when run on a class of maximum planar test

graphs Yeh has subsequently investigated practical improvements to the implementation

by Rubin

DMPHT HT and DMP b oth include paths one at a time in the plane emb edding

b eing built This observation led to DMPHT in which techniques from HT partly are used

to implement DMP

DMPHT utilize a totally dierent approach then the ones from BondyMurty and Ru

binYeh DMPHT is based on a linear Initial Phase and a Test Phase

Initial Phase The rst part of the Initial Phase is almost identical to the rst part of HT

That is two values are computed for each vertex and the adjacency lists are sorted

Then the inputgraph G is transformed to a tree T where the vertices in T represent

edgedisjoint subgraphs of G and the union of the subgraphs equals G The Initial Phase

removes the need for explicitly implementing any of the Subroutines ab ove

Test Phase A plane emb edding G is implicitly b eing built As in DMP G is initially

i i

empty then a cycle is implicitly added then one and one path is implicitly added The

inclusion of a path p through a fragment F in a region r of G will essentially only require

Lo op Determine the truth value of one or two statements and p erforming three or four

imp eratives for each new fragment emerging as a result of the the inclusion of p in G

Department of Informatics University of Oslo Norway b jornjohiuiono

Lo op Determine the truth value of one or two statements and p erforming two or four

imp eratives for each fragment F which was drawable in r

Comparison with HT DMPHT and HT seem to contain approximately the same

numb er of lines of co de

DMPHT has not b een programmed in the same language and on the same computer as

in Programmed in Simula on a Sparc DMPHT tested maximum planar graphs

with vertices in approximately seconds This is probably inferior to HT but may

b e satisfactory for practical purp oses But DMPHT has a worstcase time ratio of O n

and for sp ecial graphs unsatisfactory results may o ccur As in HT the space requirements

are of order O n

DMPHT may b e viewed as conceptually simpler then HT even though the pro of of DMP

HT is fairly long The pro of shows that a fragment F which is drawable in preferably only

one region is directly accessible that a path p through F is directly accessible from T and

that new fragments see Lo op are also directly accessible from T how new edges can b e

G thus maintaining only sorted regions and how the fact that all regions sorted added to

can b e utilized when determining in which regions new Lo op and old Lo op fragments

are drawable In DMPHT every fragment will b e determined as drawable in a maximum

of two regions and the inclusion of a path in G will result in three rather then two new

regions This will reduce the exp ected average numb er of drawable fragments in each region

thus reducing the exp ected running time of DMPHT

Also DMPHT directly tests graphs containing cutvertices and a plane emb edding is

implicitly given An expansion to explicitly give a plane emb edding in terms of regions seems

to b e straightforward This plane emb edding may also b e constructed in a static fashion

Every path included in the plane emb edding b eing built can b e emb edded p ermanently in

the plane

Testresults DMPHT was tested empirically on two classes of planar graphs

Class I Construction of a connected planar graph consisting of n n vertices First

vertices was generated Then a a cycle consisting of b etween randomly and max

random region r was rep eatedly chosen to include a vertex v in r and to generate an edge

b etween v and a vertex in the b oundary V r of r with the probability If deg v

jV r j

prevails delete all the edges incident with v and rep eatly generate edges as ab ove until

deg v Finally all the adjacency lists are to b e randomized

Class I I Maximum planar graphs were generated as ab ove with the exception that the

initial cycle contained exactly vertices and exactly new edges were generated when one

vertex v was included in a random region r

DMPHT was empirically tested on graphs generated from each of the two classes

Each of the graphs contained b etween and vertices For each graph G the

numb er of rep etitions of the interior lo ops Lo op and Lo op was counted and plotted

against jE Gj As can b e seen DMPHT b ehave in an almost linear manner for these

graphs

a Random Planar Graphs b Random Maximal Planar Graphs

L L

o o

p p

s s

Edges Edges

In the complete pap er DMPHT metho dology is presented validation of DMPHT is given

and a complete implementation of DMPHT is presented assuming that the graph to b e

tested for planarity is stored on le as a set of adjacencylists

References

J E Hop croft R Tarjan Ecient Planarity Testing J Asso c Comput Mach

K S Bo oth G S Leuker Testing the Consecutive Ones Prop erty Interval Graphs

and Graph Planarity Using PQtree algorithms J Comput Syst Sci

G Demoucron Y Malgrange R Pertuiset Graphes Planaires Reconnaissance et

Construction de Represetations Planaires Top ologiques Revue Francaise de Recherche

Operationelle Vol No st quarter

J A Bondy U S R Murty Graph Theory with Applications Macmillan London

Frank Rubin An Improved Algorithm for Testing the Planarity of a Graph IEEE

Transactions on Computers Vol c No February

Dashing Yeh Improved Planarity Algorithms Bit

A Unied Approach to Testing Emb edding and Drawing

Planar Graphs

Jo el Small

Let G V E b e a simple undirected graph with vertex set V and edge set E We rst

oer improvements to Williamsons version of the Hop croftTarjan HT planarity testing

algorithm while maintaining the On execution time Then with the data structures and

graph emb edding in place from this algorithm we augment the data structures and use

the computed emb edding to calculate and draw a rectilinear layout of the graph The

Naval Command Control and Ocean Surveilla nce Center San Diego California jsmallnoscmil

algorithms for augmenting data structures and drawing the graph run in On time as well

Furthermore the augmented data structures lend themselves to eciently generating many

nonisomorphic drawings

The planarity testing algorithm exploits the bridgecycle recursion for biconnected graphs

intro duced by HT Briey the idea is to cho ose a cycle C in G Edges in EG the edge

set of G not in EC are called the simple bridges of G with resp ect to C if b oth endp oints

lie on C Remove C together with all edges incident to C from G The resulting graph is

a collection of connected comp onents These comp onents together with the simple bridges

are called the bridges of G with resp ect to C In any plane drawing of G the cycle C will

partition the plane into two regions a nite and an innite region Each bridge must lie

entirely in one region The bridge graph of G with resp ect to C denoted BRGRGC is a

graph whose vertices are the bridges of G with resp ect to C There is an edge b etween two

vertices B and B in BRGRGC if B and B must b e drawn on opp osite sides of C in

any plane drawing of G The idea for an ecient algorithm to test if G is planar will b e to

cho ose a cycle C then construct the bridges of G with resp ect to C Add edges from C to

each bridge to make it biconnected these are called the augmented bridges and recursively

test each augmented bridge to determine if it is planar Finally compute BRGRGC and

check that this graph is bipartite If it is then this gives a valid means for lo cating the

bridges ab out C to get an emb edding of G

The bridgecycle recursion is exemplied by a data structure called the pathtree of G

denoted PATRGT Def T is a lineal spanning tree of the graph G PATRGT

is a ro oted ordered tree whose vertices are paths in G dened using T The vertices of this

tree dene a partition for b oth the vertex set VG and the edge set EG PATRGT

can b e traversed in p ostorder to test the graph for planarity We oer some improvements

to previous versions of algorithms for traversing the pathtree These improvements are

primarily in computing and maintaining the spanning forests for bridge graphs and in

simplifying the tests necessary to up date the spanning forests when adding new vertices to

the bridge graph This work has facilitated the actual implementation of the algorithms

As a part of the planarity testing vertices of the pathtree can b e colored to dene an

emb edding for the graph if it is planar or identify an obstruction if it is not If the graph

is planar we show how to augment the pathtree and use this augmented pathtree to draw

a rectilinear layout of the graph where vertices are represented by horizontal intervals and

edges are drawn using vertical intervals also called a weak visibility representation in

The edge partition dened by the pathtree is maintained in the drawing by placing edges in

the same blo ck on the same vertical line The general idea to obtain the drawing is describ ed

Supp ose we are given a planar graph G along with a PATRGT that has b een colored to

identify a planar emb edding We assign a leftright orientation to the vertices of the pathtree

that identies the orientation of the paths as they are drawn in the plane Next we can use

the PATRGT in conjunction with the orientation to assign a valid st numb ering to the

vertices Finally we compute the horizontal placement for the vertices paths of G of the

pathtree Vertices of the graph G are placed vertically according to their st numb er We

plan to describ e the algorithms to accomplish the drawings in more detail at the meeting

We have written a versatile interactive graph software package called GAP a Graph Anal

ysis Program to study graphs and graph algorithms Versions are available for b oth PCs

and SUN workstations The data structures and algorithms we describ e for testing emb ed

ding and drawing planar graphs have b een implemented as part of this package The ve

primary comp onents for doing this are to p erform a depth rst search and compute low

numb ers DFS sort the vertices using a lexicographic bucket sort LEXSORT compute

the pathtree PATR build the spanning forests for bridge graphs and test for o dd cycles

PLTEST and last augment the pathtree and compute a rectilinear layout RECDRAW

We ran the program on maximal planar graphs generated according to the following algo

rithm

pro cedure RandomGraph Graph G Numb erOfVertices nv

G K

While tv nv do

Select a face f at random

Add a new vertex v to G

tv tv

Add edges fvw g fvw g fvw g to G where

w w w are the vertices lying on f

Up date the list of faces of G

The table b elow shows the mean time in seconds for each of the comp onents on

maximal planar graphs having and vertices

jV j DFS LEXSORT PATR PLTEST RECDRAW TOTAL

BIBLIOGRAPHY

Rosenstiehl P and Tarjan R E Rectilinear Planar Layouts and Bip olar Orientations

of Planar Graphs Discrete and Computational Geometry

Tamassia R and TollisI G A Unied Approach to Visibili ty Representations of

Planar Graphs Discrete and Computational Geometry

Williamson S G Combinatorics for Computer Science Computer Science Press

Williamson S G Emb edding Graphs in the Plane Algorithmic Asp ects Annals

of Discrete Mathematics Combinatorial Mathematics Optimal Designs and Their

Applications J Srivatsava Ed North Holland

A Simple LinearTime Algorithm for Emb edding Maximal

Planar Graphs

Hermann StammWilbrandt

Intro duction

All existing algorithms for planarity testing planar embedding can b e group ed into two

principal classes Either they run in linear time but to the exp ense of complex algorithmic

concepts or complex datastructures or they are easy to understand and implement but

require more than linear time

In this pap er a new lineartime algorithm for emb edding maximal planar graphs is pro

p osed This algorithm is b oth easy to understand and easy to implement The algorithm

consists of two phases b oth of which use only simple lo cal graphmo dications

In addition to planar emb edding the new algorithm allows to test graphs for maximal

planarity We will also demonstrate how to generate random maximal planar graphs using

this algorithm The algorithm prop osed constitutes a rst step towards a simple lineartime

solution for emb edding general planar graphs A full version of this abstract may b e found

in Notice that our results make use of a seminal pap er dealing with planar bounded

orientations by Chrobak and Eppstein

One of the referees made us aware of a pap er by RC Read

Basic denitions

The terminology used in this pap er is adopted from Let G V E b e a planar graph

and AD J v for all v V denote the adjacency lists of G An embedding of G is dened

as an ordering of the adjacency lists of G such that for each vertex v V the order of

the edges in AD J v corresp onds to a counterclo ckwise traversal of the edges in any xed

embedding of G in the plane A simple planar graph is called maximal planar if the addition

of any new edge results in a nonplanar graph The smal lest maximal planar graph wrt

the numb er of vertices is dened as the complete maximal planar graph on vertices K

Thus a maximal planar graph do es not contain any vertex of degree less than This is

b ecause such graphs are triconnected The triconnectivity also enforces a unique emb edding

in the plane wrt a chosen outer face

A key concept for the new algorithm is the notion of reducible vertices

Denition A vertex v of G V E will b e called smal l if deg v otherwise it will

b e called large A vertex v V is reducible if it satises one of the following conditions a

deg v or b deg v and v has at least small neighb ors or c deg v and v

has at least small neighb ors

Institut fur Informatik I I I Universitat Bonn hermannholmiumi nformati kuni b on nde

This goal has b een recognized as a siginicant op en problem in

His metho ds are similar to those describ ed here but they need a top ological emb edding of the input graph

and they compute a straight line drawing of it in linear time and quadratic space Our algorithm runs in linear

time and linear space determines the top ological emb edding and can additionall y incorp orate Reads co ordinate

determination naturally to result in an embedding in the plane