Planar Drawings of Graphs
on the Rectilinear Grid
with Few Bends in Each Edge
research thesis
submitted in partial fulfillment
of the requirements for the degree of
master of science
in Computer Science
GILAD GRANOT
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
submitted to the senate of the technion israel institute of technology
tamuz haifa june
The work described herein was supervised by prof Shimon Even
I want to thank Shimon Even for his helpful guidance Shimon always knew to suggest
a direction to work on when I needed it He help ed me to put together my disorganized
thoughts I learned a lot from our long conversations which were fun also when not
related to the work
I want to thank Go os Kant and Arnold L Rosenberg for their useful comments on
previous versions of chapter
I thank all my friends at the Computer Science department who made my stay there
a pleasure I won t mention names so that I won t forget anyone
Finally I have to thank the various secretaries that handle graduate students I want
to thank them for not preventing me from nishing my studies
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994 i
Contents
Abstract
Background
Introduction
de nitions
Previous Work
Visibility Representations
Preliminaries
Finding a Visibility Representation of a Nonseparable Graph
Constrained Visibility Representations
Drawings with Vertices Drawn as Points
Introduction
Left Adjusted Visibility Representations
Drawing Vertices as Points
Graphs That Are Hard to Draw
Drawings with Vertices Drawn as Rectangular Mo dules
Introduction
Planarity Checking
Planar Grid Layouts The Case of No Self Lo ops
Planar Grid Layouts The Case With Self Lo ops
Mo dule Descriptions that are Hard to Draw
An Application
Non Planar Drawings
Vertices as Points
Vertices as Mo dules Grid Layouts
Bibliography
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994 ii
Abstract
For a planar graph a linear time algorithm is describ ed to construct a planar drawing
on the rectilinear grid with at most b ends in each edge The drawings found by this
algorithm for planar graphs with no self lo ops have at most b ends in each edge The
drawings use O n area and edges of length O n
We also show a planar graph that has no rectilinear planar drawing with at most
b ends in each edge and a planar graph that has no rectilinear planar drawing with at
most b end in each edge
Consider an input data which sp eci es a blo ck diagram of some mo dules and connec
tions b etween them edges which can b e drawn on the plane in such a way that lines
representing the connections do not cross The size of the mo dules and the placements of
the terminals on them is given as part of the input
A linear time algorithm is describ ed to construct a planar drawing on the rectilinear
grid with at most b ends in a line representing a connection b etween two terminals of
the same mo dule and at most b ends in any other line The external face of the drawing
may b e chosen We show a planar input with no self lo ops that has no planar drawing
with less than b ends in each edge and another planar input that has no planar drawing
with less than b ends in each self lo op
Finally we consider grid drawings of sp eci ed blo ck diagrams which are not neces
sarily planar We present a linear time algorithm to construct such grid drawings with
at most b ends in any line of the blo ck diagram We show inputs that have no such
drawing with at most b ends in any line
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Chapter
Background
Introduction
We consider drawing a graph in the plane in a way suitable for the layout of one layered
electrical circuits or VLSI comp onents The drawing is done on the rectilinear grid which
is a collection of parallel lines at a constant distance one from another and a copy of them
rotated by Edges are drawn as paths on the grid connecting their two end vertices
The drawing must b e planar that is the drawing of two di erent edges have no common
p oint except maybe at their ends where the two edges have a common end vertex
A b end is a p oint in the path representing an edge in which the path has a
angle We want the drawing to have few b ends in each edge This is imp ortant where the
drawing realizes a circuit and a b end may cause a delay or a signal loss It also makes it
easier for humans to follow the drawings of the edges We call a drawing in which every
edge has no more than k b ends a k bend drawing We investigate two types of drawings
which are di erent in the way the vertices are drawn
In the rst type of drawings we consider the vertices are drawn as distinct p oints at
intersections of grid lines A planar graph is a graph that has a planar drawing A d
planar graph is a planar graph that has no vertex of degree greater than d By de nition
drawable graphs must b e planar We describ e a linear time algorithm that pro duces a
b end drawing for any planar graph and a b end drawing for any planar graph with
no self lo ops The drawings have O n area a total of O n b ends and edges of length
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
O n We then show a planar graph that has no b end drawing and a planar graph
that has no b end drawing
In the second type of drawings we investigate the vertices are drawn as rectangular
mo dules on the grid and edges are drawn as non intersecting paths on the grid b etween
the p erimeters of two mo dules The p oint that an edge connects to the p erimeter of a
mo dule is called a terminal The size of the mo dules and the placements of the terminals
on them is given as part of the input The input assigns to each edge two sp eci c terminals
We try to pro duce drawings with few b ends in each edge We consider two versions of
the problem in one ipping of the mo dules is allowed in the other it is not Rotation of
the mo dules is always allowed
For b oth versions of the problem linear time algorithms are known that determine if a
planar drawing of the input is p ossible When we say that a planar drawing is p ossible
we do not mean that such a drawing has to b e on the rectilinear grid We will show
a linear time algorithm that constructs a b end drawing for any input that p ermits a
planar drawing In the pro duced drawings edges that are not self lo ops have at most
b ends We will show planar input with no self lo ops that has no b end drawing and
planar input that has no b end drawing
de nitions
We present here de nitions that will b e used throughout the thesis The de nitions are
gathered here so that later the explanations will b e more uent and to make them easier
to nd Some de nitions brought here may also app ear elsewhere and some de nitions
relevant only to a sp eci c chapter may only app ear there
When we talk ab out a graph G V E we are not restricted to simple graphs This
means that the graph may have parallel edges and self lo ops We denote jV j n and
jE j m For de nitions of basic terms in the context of graphs see Eve We use the
e
e
v for an edge e which connects vertices u and v and u v for a directed edge notation u
e from u to v
If a connected graph has three di erent vertices a b c such that all the paths b etween
a and c pass through b then b is called a cut vertex or a separation vertex of the graph
A connected graph that has no cut vertex is called nonseparable Nonseparable graphs
are sometimes called bi connected
A planar drawing of a graph on the plane is a mapping of it to the plane Vertices are
mapp ed to p oints or to closed curves in the plane and edges are mapp ed to curves in the
plane The endp oints of the curve representing an edge must b elong to the drawings of
the end vertices of that edge Except for this case no p oint b elonging to a drawing of an
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
edge or a vertex may b elong to a drawing of another edge or vertex
A planar graph is a graph that has a planar drawing Ad planar graph is a planar
graph that has no vertex of degree greater than d
We consider drawing a graph in the plane in a way suitable for the layout of a one
layered electrical circuit or a VLSI comp onent The drawing is done on the rectilinear
grid which is a collection of parallel lines with a constant distance b etween neighboring
lines and a copy of them rotated by Edges and vertices are drawn only on the grid
We de ne the following for drawings on the rectilinear grid A bend is a p oint in the
path representing an edge in which the path has a angle We call a drawing in which
every edge has no more than k b ends a k bend drawing The area of a drawing is the area
of the smallest rectangle sides parallel to the grid lines containing the drawing The area
is measured in grid squares
When we remove from the plane p oints that b elong to a planar drawing of a connected
graph the plane is divided into continuous areas Points inside a drawing of a vertex are
also removed The b oundary of each of these areas represents a cycle of the graph an
edge may app ear twice in such a cycle Theses cycles are called the faces of the drawing
The face which represents the b oundary of the in nite area is called the external face
An edge incidence ordering of a graph is a sp eci cation for each vertex of a cyclic
order of the endp oints of its incident edges A graph with an edge ordering is called
an ordered graph A drawing of an ordered graph in the plane is proper if the edge
ordering of the ordered graph corresp onds to the clo ckwise arrangement of the endp oints
around the vertices in the drawing A planar ordered graph is an ordered graph that has
a prop er planar drawing The edge ordering of a planar ordered graph is called a planar
representation An ordered graph determines the faces of any prop er drawing but do esn t
determine which one is external
An st bipolar orientation of a graph G is a directed acyclic graph whose underlying
undirected graph is G it has one source s one sink t and an edge s t
In an st bip olar orientation of a planar ordered graph the only source sink of a
subgraph consisting of one face is called the source sink of the face As will b e shown
later Lemma in every st bip olar orientation of a planar ordered graph for every ver
tex the incoming outgoing edges app ear consecutively The set of consecutive incoming
outgoing edges of a vertex is called an edge block
A visibility representation of a graph is a planar drawing of the graph on the rectilinear
grid where vertices are drawn as horizontal lines or p oints and edges are drawn as vertical
lines A visibility representation of a graph induces directions to the edges of the graph
the edges are directed from the lower vertex to the higher vertex We assume that the ends of the drawing of a vertex are at the endp oints of it s leftmost and rightmost incident
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
edges
We will call a vertex in a visibility representation left adjusted if the leftmost p oint of
its drawing is the end of the drawings of two edges An LAVR Left Adjusted Visibility
Representation of a graph is a visibility representation such that in the directed graph it
induces every vertex which isn t left adjusted is a source or a sink See gure for an
example
Previous Work
Drawings on the rectilinear grid have b een studied b ecause of their applications in VLSI
planning and automatic graph and data drawing DETT TDB Planar drawings
in which lines do not cross have b een given a sp ecial consideration In planar drawings
following the drawings of edges is easier and the drawing can b e used as a layout of a
one layered electrical circuit
Most of the previous work dealt with drawings in which vertices are drawn as grid
p oints Several asp ects of such drawings have b een considered The area of the drawing is
the area of the smallest rectangle sides parallel to the grid lines containing the drawing
The area is measured in grid squares and is considered b etter when small It is known that
every planar graph can b e drawn in O n area Shi TT and that some planar
graphs require n area to b e drawn Shi Other prop erties of drawings which are
of interest are the total edge length the maximum length of an edge MO and the
total number of b ends Sto Tam TT TTV A b end is a p oint in the path
representing an edge in which the path has a angle
Our interest is in drawings which minimize b ends but instead of considering the total
number of b ends in the drawing we want to minimize the maximum number of b ends
which o ccur in any edge This is imp ortant where the drawing realizes a circuit and a b end
may cause delay or signal loss It also makes it easier for humans to follow the drawings
of the edges We call a drawing in which every edge has no more than k b ends a k bend
drawing See gure An algorithm in Tam given k and a planar representation
nds a k b end drawing consistent with the representation with the minimum total number
of b ends in O n log n time In TT a linear time algorithm is given to nd a b end
drawing for simple graphs Kant Kan shows that every planar graph without self
lo ops and with ve vertices or more has a b end drawing however the choice of the
external face is determined by the algorithm Two relevant pap ers have b een written
after our do cumentation of the results of chapter they are
Biedl Bie shows a linear time algorithm to nd a b end nonplanar drawing of
a graph
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Biedl and Kant BK show a linear time algorithm to nd b end planar drawings
of most simple graphs
So far we have only referred to drawings in which the vertices are drawn as grid
p oints In chapters we consider drawings in which vertices are drawn as rectangles on
the grid The shap e of each rectangle and the placements of the edge endp oints on its
p erimeter are given as part of the input Although such drawings seem natural for VLSI
planning we do not know of algorithms for nding such planar drawings We only know of algorithms to check if a planar drawing exists for a given input Pin Ami Zak
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Chapter
Visibil i ty Representations
In this chapter we describ e known algorithms for nding visibility representations of
graphs The algorithms are describ ed for two reasons The rst reason is that the the
descriptions of the algorithms makes the thesis more self contained And the second
reason is to clarify the use of these algorithms in later chapters
Pro ofs are omitted in this chapter since they can b e found in the referenced work
Preliminaries
An edge incidence ordering of a graph is a sp eci cation for each vertex of a cyclic
order of the endp oints of its incident edges A graph with an edge ordering is called
an ordered graph A drawing of an ordered graph in the plane is proper if the edge
ordering of the ordered graph corresp onds to the clo ckwise arrangement of the endp oints
around the vertices in the drawing A planar ordered graph is an ordered graph that has
a prop er planar drawing The edge ordering of a planar ordered graph is called a planar
representation An ordered graph determines the faces of any prop er drawing but do esn t
determine which one will b e external
Graphs are usually represented in computers by adjacency lists The order of the
incident edges of a vertex in the list can b e viewed as their cyclic order Therefore
ordered graphs can b e represented in a computer in the same way that regular graphs
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
are When dealing with planar graphs we will usually op erate on a corresp onding planar
ordered graph
An st bipolar orientation of a graph G is a directed acyclic graph whose underlying
undirected graph is G it has one source s one sink t and an edge s t Given a non
t an st bip olar orientation of the graph with separable graph and one of its edges s
the sp eci ed s t edge can b e found in linear time ET Even and Tarjan use an
equivalent notion of an st numbering
Lemma RT TT In an st bipolar orientation of a planar ordered graph the
fol lowing hold
For every vertex its incoming outgoing edges appear consecutively in the edge
ordering
Every face consists of two edge disjoint directed paths
Thus if we take a subgraph consisting of one face of an st bip olar orientation of a planar
ordered graph the subgraph has exactly one source and one sink called the source and
the sink of the face The set of consecutive incoming outgoing edges of a vertex is called
an edge block
Let us consider an st bip olar orientation whose edges have non negative integer weights
The weight of the longest path from s to each vertex v which we denote by d v can
b e found in linear time See for example Eve pp A top ological ordering
for a directed acyclic graph with non negative edge weights is an assignment of a p ositive
integer to each vertex such that for each edge the number assigned to its destination
minus the number assigned to its origin is greater than or equal to the weight of the edge
The values of d are a top ological ordering
We lo ok at an edge e of a directed graph with a planar representation We will denote
the face to the left right of e when e is drawn upwards as L e R e
A visibility representation of a graph is a planar drawing of the graph on the rectilinear
grid where vertices are drawn as horizontal lines or p oints and edges are drawn as vertical
lines A visibility representation of a graph induces directions to the edges of the graph
the edges are directed from the lower vertex to the higher vertex We assume that the
ends of the drawing of a vertex are at the endp oints of it s leftmost and rightmost incident
edges A visibility representation is describ ed by the following data For each vertex v
its Y axis co ordinate y v and for each edge e its X axis co ordinate x e
We will call a vertex in a visibility representation left adjusted if the leftmost p oint of
its drawing is the end of the drawings of two edges An LAVR Left Adjusted Visibility
Representation of a graph is a visibility representation such that in the directed graph it
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
induces every vertex which is not left adjusted is a source or a sink See gure for an
example
We will now describ e two known linear time algorithms for nding visibility represen
tations The di erence b etween the two algorithms is that the second algorithm nds a
visibility representation with some constraints
Figure a graph and an LAVR representation of it
åìù ìàîùì øùåéî éåå÷ øåéöå óøâ :2.1 íéùøú
Finding a Visibility Representation of a Non
separable Graph
In this section we describ e an algorithm of Rosenstiehl and Tarjan RT that was
discovered indep endently by Tamassia and Tollis TT
The directed dual graph G V E of an st bip olar orientation G V E with a planar
representation is de ned as follows
V ff j f is a face of Gg
E fL e R e j e E fs tg g fR s t L s t g
It can b e shown that G is an st bip olar orientation with R s t L s t b eing the
s t edge
Theorem RT TT algorithm VR described in gure nds a visibility rep
resentation for any non separable graph without self loops in linear time
The s t edge is the leftmost edge in the visibility representation and the face on its
left is the external face Therefore the choice of s t in the algorithm ensures that f will
b e the external face
The edges in the visibility representation are drawn upward according to their direction
in the st bip olar orientation of G
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Lemma The visibility representations produced by algorithm VR are left adjusted
Proof Let v s t and let e b e the leftmost incoming edge of v and e b e the
leftmost outgoing edge of v Since d L e d L e x e x e
The height of the drawing is less than jV j and the width of the drawing is less than
the number of faces of G
pro cedure VR G f f RT very similar to TT g
f G is a planar undirected nonseparable ordered graph and its data structure g
f speci es its planar representation The face f is speci ed to be external g
b egin
let s b e a vertex on f
e
let s t b e an edge in the clo ckwise traversal of f
direct the edges of G as an s t bip olar orientation f ET g
nd G the directed dual graph of G
set all edge weights to
calculate d v for the vertices of G and d v for the vertices of G
for each vertex v of G do
y v d v
e
x s t
e
0
for each edge e E fs tg do
0 0
x e d L e
end
Figure nding an LAVR of a nonseparable graph
÷éøô-éà óøâ ìù ìàîùì øùåéî éåå÷ øåéö úàéöî :2.2 íéùøú
As will b e shown later the VR algorithm can b e used to construct a visibility rep
resentation of any planar connected graph with no self lo ops If the graph is connected
but separable it can b e made nonseparable by adding edges These edges will b e called
auxiliary An LAVR of the graph with the auxiliary edges can b e found using the VR
algorithm After removing the drawings of the auxiliary edges from the resulting LAVR
one gets a visibility representation of the graph Notice that this visibility representation
may not b e left adjusted An algorithm for nding an LAVR of a connected graph is
describ ed in section
Constrained Visibility Representations
In this section we are interested in pro ducing visibility representations under certain
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
constraints we want certain paths of the graph to b e drawn as straight vertical lines We
describ e an Algorithm of Di Battista Tamassia and Tollis DTT that runs in linear
time For consistency with the rest of the thesis we describ e this algorithm in a slightly
di erent way than in DTT
The input consists of a planar ordered graph which is an st bip olar orientation G V E
and some constraints The output is a visibility representation of G that is prop er in
which all edges are drawn p ointing upwards What we have describ ed so far can b e done
by the VR algorithm if one removes the nding of the st bip olar orientation from the
algorithm Next we describ e the constraints imp osed on the visibility representation
which force us to use the new algorithm
A set of mutually edge disjoint directed paths of the st bip olar orientation is
sp eci ed in the input All the sp eci ed directed paths have to b e aligned in the output
visibility representation That is the X axis co ordinate of all edges b elonging to the
same path must b e the same The sp eci ed paths may not cross one another in a prop er
drawing That is for any four edges e e e e incident to the same vertex in this order
not necessarily consecutively if e and e b elong to the same path then e and e may
not b elong to the same path
If some edge do es not app ear in a path of it can b e added as new path to
Therefore we may assume that every edge app ears in one path of We denote by p e
the path of that an edge e E app ears in
We de ne the constrained dual graph G V E of G V E as follows
V ff j f is a face of Gg
E fp e R e j e E g
w
E fL e p e j e E fs tg g fp s t L s t g
z
E E E
w z
It can b e shown that G is an st bip olar orientation with p s t L s t b eing the s t
edge
Theorem DTT algorithm CVR that is described in gure nds a constrained
visibility representation for any st bipolar orientation of a planar ordered graph in linear
time
If one applies CVR G but assigns higher integer weights to the edges the result
is still a constrained visibility representation This can b e used to creating gaps in the visibility representation in sp eci c places
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
pro cedure CVR G f DTT very similar to TT g
f G V E is an st bipolar orientation of a planar ordered graph g
f is the set of aligned paths g
b egin
Construct G V E E the constrained dual graph of G
w r
Set edge weights as follows e E w e
e E w e e E w e
w z
Calculate d v d v for the vertices of G G resp ectively
for each vertex v of G do
y v d v
for each edge e E do
x e d p e
end
Figure nding a constrained visibility representation
ìáâåî éåå÷ øåéö úàéöî :2.3 íéùøú
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Chapter
Drawings with Vertices Drawn as
Points
Introduction
In this chapter we consider drawing a planar graph on the rectilinear grid with no
crossings such that there are few b ends in the drawing of every edge Vertices are drawn
as p oints at intersections of grid lines Edges are drawn as non intersecting paths on the
grid By de nition drawable graphs are planar
In this chapter we denote G as the underlying undirected graph of the directed graph
G
Planar drawings on the rectilinear grid have b een studied b ecause of their applications
in VLSI planning and automatic graph and data drawing TDB Several asp ects of
the drawings have b een considered The area of the drawing is the area of the smallest
rectangle sides parallel to the grid lines containing the drawing The area is measured in
grid squares and is considered b etter when small It is known that every planar graph
can b e drawn in O n area Shi TT and that some planar graphs require n
area to b e drawn Shi Other prop erties of drawings which are of interest are the total
edge length the maximum length of an edge MO and the total number of b ends
Sto Tam TT TTV A b end is a p oint in the path representing an edge in
which the path has a angle
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Our interest is in drawings which minimize b ends but instead of considering the total
number of b ends in the drawing we want to minimize the maximum number of b ends
which o ccur in any edge This is imp ortant where the drawing realizes a circuit and a b end
may cause delay or signal loss It also makes it easier for humans to follow the drawings
of the edges We call a drawing in which every edge has no more than k b ends a k bend
drawing See gure An algorithm in Tam given k and a planar representation
nds a k b end drawing consistent with the representation with the minimum total number
of b ends in O n log n time In TT a linear time algorithm is given to nd a b end
drawing for simple graphs Kant Kan shows that every planar graph without self
lo ops and with ve vertices or more has a b end drawing However the choice of the
external face is determined by the algorithm
Figure a b end drawing which is not a b end drawing
éúééðô-2 àì àåäù éúééðô-3 øåéö :3.1 íéùøú
We describ e an algorithm that pro duces b end drawings for planar graphs and
b end drawings for planar graphs The drawings have O n area a total of O n b ends
and edges of length O n We then show a planar graph that has no b end drawing
and a planar graph that has no b end drawing The algorithm runs in linear time
We will draw each connected comp onent of the graph separately and thus consider
from now on only connected graphs
The algorithm starts by nding an LAVR of the graph without its self lo ops Next
the drawing is converted to a drawing with vertices drawn as p oints
Left Adjusted Visibility Representations
We start with some preliminaries
The input to our drawing algorithm is a planar ordered graph and a sp eci cation of
the face which is to b e external If no external face is sp eci ed we can choose any face
to b e external The output is a drawing congruous with the input data
Linear time algorithms for checking the planarity of a graph and nding a planar
representation of it are known LEC ET BL CNAO HT Therefore if we
do not have a planar representation of the graph or do not know if the graph is planar
then we can check for planarity and nd a planar representation while keeping the running
time linear We will work with planar ordered graphs and assume that the data structure
of the graph sp eci es its planar representation
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
We assume knowledge of the discussion of st bip olar orientations and the related
de nitions that app ear in section
Lemma A graph that has an st bipolar orientation is nonseparable
We denote the input graph without its self lo ops G Recall that the planar repre
sentation and chosen external face of G are derived from the input data The drawing
algorithm of this chapter uses an LAVR of G We show an algorithm that given G a
connected ordered graph with no self lo ops which may b e separable and f a face of G
nds an LAVR of G that is prop er and f is its external face This algorithm generalizes
the algorithm in RT describ ed in section to nd an LAVR of connected graphs
which are not necessarily nonseparable
0
Our algorithm mo di es G V E to form a planar ordered graph G V E G s
00
undirected underlying version G V E contains G as a subgraph and is nonseparable
00
The edges of G which corresp ond to edges in E E are called auxiliary If we remove
the auxiliary edges from G the planar representation is the same as that of G The con
struction of G from G is done as follows The edges of G are directed one nonseparable
comp onent after another Each comp onent is directed as an st bip olar orientation The
auxiliary edges are comp osed of up to one auxiliary edge for each directed comp onent
each auxiliary edge corresp onds to a di erent comp onent After directing the edges of
each comp onent the comp onents directed so far together with the auxiliary edges corre
sp onding to them constitute an st bip olar orientation we will call this temp orary graph
0
G The auxiliary edge corresp onding to each comp onent is determined at the time the
comp onent is directed but it is only added later The auxiliary edges are added only
after directing all of the edges When directing a comp onent C other than the rst that is
0
connected to G by a cut vertex u it is directed in one of two ways dep ending on u b eing
0
the sink of the face of G containing C see gure The auxiliary edge corresp onding
to C is also shown in gure However in case b of gure this edge may not b e
eventually added to the graph if it is not needed in this case there will b e no auxiliary
edge corresp onding to C in G
u
case a case b
t
e
new
comp onent
t
new
comp onent
s
s
u
e
e is the auxiliary edge corresp onding to the new comp onent
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Figure a visual description of the two cases of directing a nonseparable comp onent
÷éøô-éà áéëø ìù úåúù÷ì ïååë ïúî ìù íéø÷îä éðù ìù éìàåæéå øåàéú :3.2 íéùøú
Next the auxiliary edges are added to the graph This is done according to markings
made when directing the graph Following an LAVR of the new graph is found by the
algorithm in RT We remove the auxiliary edges from the visibility representation and
it remains an LAVR due to the sp ecial way the auxiliary edges have b een added Our
algorithm is describ ed in gures
Remarks
We p erform DFS on G to nd the nonseparable comp onents We start the DFS
e
t and choose the edges incident to a vertex in the order of their at the st edge s
r r
placement in the planar representation starting with the edge through which the
vertex has b een reached Later when we do a DFS on the nonseparable comp onents
tree we traverse the comp onents in the order in which their rst edge is traversed
here This ensures that we won t traverse a comp onent that is inside a face of
another yet untraversed comp onent
The rst edge in the clo ckwise traversal of an external face of a comp onent starting
at the entry cut vertex of the comp onent can b e found easily It is the rst traversed
edge of the comp onent in the DFS used to determine the nonseparable comp onents
See remark
00
Observe that the edges of f are a subset of the edges of the face of G containing
00
f but some of the edges of this face may not b e directed yet These latter edges
b elong to the external faces of other comp onents of G
If a vertex is not a source or a sink and if its blo ck of incoming outgoing edges
consists of more than one edge then the rst and last edges of the blo ck are called
extreme The remaining visibility representation after the removal of the auxiliary edges
is ensured to b e left adjusted by making sure that no auxiliary edge is extreme
Lemma The auxiliary edges in G as constructed by the algorithm are not extreme
Proof After the rst edge adding stage each of the added auxiliary edges is the only
00 0
incoming edge of its end vertex s In the edge blo cks of their start vertex s these
edges are inserted b etween the two edges of f Therefore after the rst stage there are
no extreme auxiliary edges
After the second edge adding stage each of the added auxiliary edges is the only
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
00 0
outgoing edge of its start vertex t In the edge blo cks of their end vertices t they
are always inserted b etween the two edges of f Therefore the edges added in this stage
are not extreme What is left to show is that the edges that are added in the rst stage
do not b ecome extreme in the second stage This may happ en if an edge is added to an
edge blo ck that contained only one auxiliary edge For this to happ en an incoming edge
has to b e added to a vertex marked as a temp orary source In the second stage we add
only incoming edges to vertices marked as sinks of faces A vertex marked as a temp orary
0
pro cedure LAVR G f
0
f G V E is a planar undirected ordered graph The face f is speci ed as external g
b egin
0
let s b e a vertex on f
r
e
0
let s t b e an edge in the clo ckwise traversal of f
r r
e
orientation G s t f see gure g
r r
f The current graph is G a directed version of G g
f Every face of G has two vertices marked as its source and sink g
f Also vertices may be marked as temporary sources or as temporary sinks g
for every face f of G do f rst adding stage g
b egin
0
let s b e the vertex marked as the source f
0
e
0 0
traverse f clo ckwise starting at s with s a
00 0
for every vertex s traversed that is marked as a temp orary source and isn t s do
0 00
b egin add a new auxiliary edge s s inside f end
0 0
f The added edges are ordered consecutively around s after e in their creation order g
the vertex marked as the sink of f is marked as the sink of all the newly created faces
end
f Now the faces of the current graph may be di erent g
for every face f of the current graph do f second adding stage g
b egin
0
let t b e the vertex marked as the sink of f
0
e
0 0
a traverse f clo ckwise starting at t with t
00 0
for every vertex t traversed that is marked as a temp orary sink and isn t t do
00 0
b egin add a new auxiliary edge t t inside f end
0 0
f The added edges are ordered consecutively around t after e in their creation order g
end
f We call the resulting nonseparable graph G g
nd an LAVR of G such that s t is the leftmost edge f RT section g
r r
remove the auxiliary edges from the LAVR of G to pro duce an LAVR of G
end
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Figure nding an LAVR of a connected graph
øéù÷ óøâ ìù ìàîùì øùåéî éåå÷ øåéö úàéöî :3.3 íéùøú
e
pro cedure orientation G s t
r r
f a subroutine of the algorithm in gure that directs the edges of G to produce G g
b egin
nd the nonseparable comp onents of G f see remark g
let c b e the comp onent containing e
r
nd the nonseparable comp onents tree of G such that the ro ot comp onent is c
r
e
direct the edges of c as an st bip olar orientation with s t b eing the s t edge f ET g
r r r
00
mark the source sink of each face f of c as the source sink
r
00
of the face of G containing f f see remark g
do a DFS on the comp onent tree and when rst reaching a comp onent c other than c do
r
b egin f see remark g
0
let s b e the entry cut vertex of the comp onent in the tree
let f b e the external face of c
0 0 0
t b e the rst edge in the clo ckwise traversal of f starting at s f see remark g let s
0
if s is the vertex marked as the sink of the face of G that f is a part of then
b egin
0 0
direct the edges of c as an st bip olar orientation with t s b eing the s t edge
0
mark t as a temp orary source
end
else
b egin
0 0
direct the edges of c as an st bip olar orientation with s t b eing the s t edge
0
mark t as a temp orary sink
0
if s is marked as a temp orary sink then unmark it
end
00
mark the source sink of each face f of c other than f as the source sink
00
of the face of G containing f f see remark g
end end
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Figure a subroutine of the algorithm in gure
3.3 íéùøúá íúéøåâìàä ìù äøâéù :3.4 íéùøú
The dotted lines represent parts of the drawing that are removed due to the removal
of the auxiliary edges
Figure input and output of the LAVR algorithm
ìàîùì øùåéî éåå÷ øåéö úàéöîì íúéøåâìàä ìù èìôäå èì÷ä :3.5 íéùøú
source cannot b e marked as a sink of a face therefore G has no extreme auxiliary edges
Lemma G the undirected underlying version of G is a nonseparable planar or
dered graph
Proof G is planar and the auxiliary edges are added inside faces of the current graph
without introducing edge crossings Therefore G is a planar ordered graph
e
Next we show that G is an st bip olar orientation with s t b eing the s t edge We
r r
have to show that s and t are the only source and sink of G and that G is acyclic
All the sources and sinks of G other than s t are marked as temp orary sources and
temp orary sinks resp ectively Following we show that the vertices marked as temp orary
sources and sinks are not sources and sinks of G and therefore the only source and sink
of G are s and t
r r
Every vertex v that is marked as a temp orary source app ears on a face f of G for
which v is not marked as a source This is the face containing the external face of the
nonseparable comp onent closest to the ro ot in the comp onent tree that contains v In
the rst adding stage while traversing f an edge entering v is added Therefore v is not
a source of G
Every vertex v that is marked as a temp orary sink app ears on a face f of G for
which v is not marked as a sink If f is split to several faces in the rst adding stage the
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
resulting face that contains v still has the same marked source which is not v Therefore
in the second adding stage an edge leaving v is added Consequently v is not a sink of
G
Now we show that G has no directed cycles First we show that the faces of G are
not directed cycles and then we show that this implies that G has no directed cycles
Clearly the faces of G are not directed cycles Observe that after the rst adding
stage there are no faces that are directed cycles this is b ecause the new faces that are
formed by adding auxiliary edges have a source which is the start vertex of the auxiliary
edge After the second adding stage there are still no faces that are directed cycles since
the newly formed faces have a sink which is the end vertex of the new auxiliary edge So
we see that the faces of G are not directed cycles
We prove that G has no directed cycle by contradiction Let s assume that G has a
directed cycle We showed that G is planar We lo ok at a prop er planar drawing of G
in which the s t edge is on the external face Every cycle in G b ounds a nite area in
the drawing We take a directed cycle c that b ounds the least area in the drawing The s
and t vertices are outside of the area c b ounds since they can t b e on c and the s t edge
is on the external face This implies that there is no source or sink inside the b ounded
area There must b e an edge incident to a vertex on c that is drawn in the b ounded
area otherwise c is a face and we have seen that a face can t b e a directed cycle We
trace a path in the b ounded area starting with this edge going through edges in the same
direction until we reach a vertex which is on c This must happ en since there is neither
a source nor a sink nor a smaller directed cycle in the b ounded area This path with
a part of c constitutes a smaller directed cycle A contradiction
Thus G has no directed cycles and has only one source and one sink connected by
an edge rendering it an st bip olar orientation By Lemma G is nonseparable
Theorem The algorithm in gure nds an LAVR for any connected planar or
dered graph with no self loops Also the drawing is proper and has the given external
face
Proof G is a subgraph of G The planar representation of the original edges is the
same We have shown in Lemma that G is nonseparable and planar and therefore
we can nd an LAVR of it RT After deleting the auxiliary edges we get a visibility
representation of the original graph which is prop er
The side of the s t edge which is in the given external face of G is drawn in the
external face of the LAVR of G After the deletion of the auxiliary edges it stays in the
external face of the drawing Therefore the given external face which contains this side
of the s t edge is the external face of the drawing of G
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
What is left to b e shown is that the visibility representation of G that is pro duced is
left adjusted The edges that are removed the auxiliary edges are not extreme as shown
in Lemma Therefore the leftmost edge of an edge blo ck is not removed unless the
edge blo ck contains only this edge or the vertex of the edge blo ck is a source or a sink
of G This means that a vertex that has b oth outgoing and incoming edges in G will
either keep b oth its leftmost incoming and outgoing edges or it will b ecome a source or
a sink of G Therefore the drawing remains an LAVR after the deletion of the auxiliary
edges
Clearly the algorithm for nding an LAVR runs in linear time including the algorithm
used from RT which runs in linear time see section As we will see in the following
section changing the LAVR to the wanted drawing can also b e done in linear time Thus
the whole pro cedure runs in linear time
Let us consider the area of the determined LAVR We get an LAVR for G as in
0 0 0
RT We denote the number of faces edges and vertices of G by f m n resp ectively
According to the algorithm of RT see section the height of the LAVR is at most
0 0
n n and its width is at most f The original graph is planar so we
0
have m n We add at most n auxiliary edges so we have m n By Euler s
0 0 0
formula we have f m n n n n Therefore the width of
the LAVR is at most n and its height is at most n
Drawing Vertices as Points
In this section we consider the construction of a planar drawing with vertices drawn as
p oints
A self lo op adjacent to a vertex of degree four whose endp oints do not app ear con
secutively around the vertex can not b e drawn with less than b ends These incidences
can b e searched for and corrected by changing the planar representation at such vertices
in linear time It is the the only exception in which the planar representation has to b e
changed
If the graph has only one vertex and two self lo ops then we will draw it in the obvious
way A problem arises if a face containing only one of the self lo ops is supp osed to b e
the external face There is no b end drawing of this case in fact a b end drawing is
required This is the only exception to the free choice of the external face
If the connected graph has more than one vertex then we will use a pro cedure similar
to the one used in TT to create the desired drawing We nd an LAVR of the graph
without its self lo ops using the algorithm in gure Next we change the drawing so
that vertices are drawn as p oints This is done by applying transformations to the vertices
A transformation on a vertex in the LAVR transforms the shap e of the vertex and the
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
connections of its incident edges It also adds a self lo op to the vertex if one existed in the
original graph The shap es of edges change just near their ends the rest remains a vertical
line The transformations are describ ed in gure Some transformations require new
grid lines to b e added to the drawing These are inserted to the drawing so that all the
lines in the new shap e of the vertex can b e drawn on grid lines This increases the area of
the drawing The height and width of the drawing are increased with each transformation
by at most two Therefore the height and width of the drawing increase by no more than
The horizontal p osition of the edges n each and the area of the drawing remains O n
after the transformations stays the same except for the comp ensations for the added
grid lines The nal drawing can b e found in linear time
Another approach to the pro duction of the nal drawing is to mo dify the transfor
mations to record only the shap e of the drawing but omit the length of the lines This
pro cedure pro duces an orthogonal representation TT The orthogonal representation
can b e converted to the nal drawing in linear time Tam This approach may lead
to a smaller multiplicative constant in the b ound on the total number of b ends in the
drawing but it also leads to a bigger constant in the b ound on the area
T1 T13 T7 T19 T2 T14
T8 T15 T20 T3
T16 T21 T9 T4 T22
T10 T17 T23 T5 T11 T18 T24
T6 T12
Figure transformations on the drawings of vertices in an LAVR
ìàîùì øùåéî éåå÷ ÷åéöá úåúù÷ä øåéö ìò úåéöîøåôñðøè :3.6 íéùøú
The op eration of the whole drawing pro cedure is illustrated in gure The graph
drawn is taken from TTV
Theorem After performing the transformations every edge has no more than
bends
Proof In the pro of we will lo ok at the edges of G as directed The directions induced by
the visibility representation of G are the same as in G All the transformations except
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
for T T present no more than two b ends at the b ottom of an edge and no more than
one b end at the top of an edge Therefore every edge that isn t the leftmost incoming
edge of a vertex on which T or T has b een p erformed has no more than three b ends
What is left to prove is that those edges which have two b ends at the top have no
e
0 00
more than one b end at the b ottom Such an edge v v is the s t edge of the st bip olar
orientation of its comp onent c since it is the leftmost incoming edge of the sink of c This
0
among those which b elong to c If e is implies that e is the leftmost outgoing edge of v (a) (c)
(b)
a a graph b an LAVR of the graph c a b end drawing of the graph
Figure a demonstration of the algorithm
íúéøåâìàä ìù äîâãä :3.7 íéùøú
0
not the rightmost outgoing edge of v then it has no more than one b end in the b ottom
0
Otherwise e is the only outgoing edge of v b elonging to c so c is comp osed only of e This
00
means that there are two comp onents connected at v and they are b oth in the external
e
00 00 0
v is s t So face of the other This is p ossible only if v is t which implies that v
r r r
e is the leftmost outgoing edge of v which is a source and therefore e has no more than
one b end at the b ottom
It is imp ortant that we use an LAVR since the transformations work only on left
adjusted vertices Otherwise we could have had vertices with shap es for which suitable
transformations don t exist
The transformations of vertices without self lo ops present at most four b ends at each
vertex Therefore the total number of b ends is at most n In case of a graph with self
lo ops two of the transformations present six b ends thus increasing the upp er b ound on
the total number of b ends The edges remain of length O n
When the algorithm is applied to a planar graph with no self lo ops the transfor
mations present at most one b end at each end of every edge Thus a b end drawing is
pro duced for planar graphs For planar graphs one can transform the LAVR into a
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
rectilinear drawing by simply choosing a p oint to represent each vertex Obviously the
area stays the same as in the LAVR
Here is an example of a vertex with a shap e that can t app ear in an LAVR for which
1
there is no transformation that presents no more than two b ends at the b ottom of
an edge and no more than one b end at the top of an edge
Graphs That Are Hard to Draw
We show here graphs that are hard to draw These graphs are not ordered graphs
Following we prove that there isn t a b end b end drawing for every planar
planar graph
Lemma The planar and triangulated simple graph in which every vertex is of degree
four octahedron see gure has no bend drawing
Proof This graph is triconnected and therefore has only one planar representation
which uniquely determines the faces We choose any face to b e the external face and
show that it can t b e drawn with no more than two b ends in each edge in a way that the
rest of the graph can b e drawn inside
The external face as all the others has three edges and its three vertices are of degree
four In the drawing of the external face the drawing of each vertex must have a angle
and a angle where the angle must b e inside otherwise the two edges that are
incident to the vertex and are not in the external face can t b e connected to the vertex
We show that such a drawing of the external face is not p ossible with at most two b ends
in each edge
When we go around the drawing of the face which is a closed curve at the drawing
of each vertex we turn towards the outside of the curve To close the curve we must
turn towards the inside of the curve and to comp ensate for the turns at the vertices
we must turn inside Each b end on an edge can turn at most inside so we will
b ends to draw the face With at most two b ends in each of the three need at least
edges there can t b e seven b ends in the external face
o ctahedron tetrahedron
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Figure the graphs used in lemmas and
. 3.4,3.5 úåîìá ùåîéùáù íéôøâä :3.8 íéùøú
Lemma The planar and triangulated simple graph in which every vertex is of degree
three tetrahedron see gure has no bend drawing
Proof The pro of is similar to the previous lemma Again we show we can t draw the
external face in a way that the rest of the graph can b e drawn Here at every vertex at
b est we turn so we have to turn inside with the b ends This requires four b ends
but with at most one b end in each of the three edges there can t b e four b ends in the
external face
Lemmas and imply that for k there is a k planar graph for which a
k b end drawing do es not exist The case for k is trivial
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Chapter
Drawings with Vertices Drawn as
Rectangular Mo dules
Introduction
In this chapter we consider drawings in which the vertices are drawn as rectangular
mo dules on the grid that is blo ck diagrams Not much work has b een done on drawings
in which vertices are more complex structures in spite of the fact that this approach is
natural in VLSI where the vertices can represent prede ned mo dules
As b efore our interest is in drawings which have a small constant upp er b ound on the
number of b ends which o ccur in any edge This is imp ortant where the drawing realizes
a circuit and a b end may cause delay or signal loss
We want to control certain qualities of the blo ck diagram that we draw nd a layout
of Therefore more data is sp eci ed in the input than just a graph
The input for drawing a blo ck diagram called a module description consists of
A set of rectangular mo dules of integral dimensions Each mo dule has a set of
terminals which are designated p oints on the mo dule s p erimeter at integral places
A set of connections Each connection has two designated terminals which are its
end p oints and each terminal is an end p oint of exactly one connection
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
The task of drawing the blo ck diagram on the rectilinear planar grid consists of
Placing each mo dule on the grid adjusted to the grid lines The placed mo dules
are nonoverlapping and do not touch each other
Drawing each connection as a path on the grid so that
No two such paths share a grid edge
If two paths share a grid p oint then they cross each other at that p oint
No path shares any p oint with any mo dule except its designated end p oints
Such a drawing is referred to as a grid layout of the mo dule description If no two
paths intersect the grid layout is called planar See gure
Figure a b end planar grid layout which is not b end
úéúééðô-2 àì àéäù úéúééðô-3 úéøåùéî úéâéøñ äùéøô :4.1 íéùøú
Clearly a mo dule description corresp onds to an ordered graph Each mo dule is rep
resented by a vertex and each connection is represented by an edge of the graph Note
that the ordered graph is not necessarily simple it may have parallel edges and self lo ops
If this ordered graph is planar we say that the mo dule description is planar A mo dule
description has a planar grid layout without ippings if and only if it is planar
When referring to a mo dule description we will frequently use terms b elonging to
the corresp onding ordered graph That is we will say edge and degree of a vertex in
stead of connection and number of terminals on a mo dule We denote the ordered graph
corresp onding to the input mo dule description as G V E
There are several natural tasks concerning grid layout They can b e classi ed accord
ing to the following issues
Do we insist on a planar grid layout If we do the corresp onding ordered graph
must b e planar
Do we allow to ip mo dules Of course ipping a mo dule reverses the edge ordering
in the corresp onding vertex
There are known linear time algorithms to check if a given ordered graph is planar
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Also in case ipping is allowed there is a known linear time algorithm to check if a
ipping exists for which the resulting ordered graph is planar and if so to sp ecify such
a ipping Applying these ippings to the original mo dule description yields a planar
mo dule description In such a case the mo dule description is called ip planar We allow
rotation of the mo dules since it do es not e ect planarity
Here b ends have the same meaning as b efore A bend is a p oint of a path in a grid
turn we say that the corresp onding edge of the layout in which the path makes a
ordered graph has a b end A grid layout is called k bend if no edge path has more than
k b ends
In this chapter we present two linear time algorithms
For any given planar mo dule description as well as a sp eci ed external face a
b end planar grid layout is constructed Each edge which is not a self lo op has
at most b ends This result is the b est p ossible in the following sense There
are ip planar mo dule descriptions with self lo ops for which none of their planar
mo dule descriptions after ippings has a b end planar grid layout Also there
are ip planar mo dule descriptions without self lo ops for which none of their planar
mo dule descriptions has a b end planar grid layout
For any mo dule description a b end grid layout is constructed It may b e nonpla
nar There is no need to rotate or ip any mo dule Again this result is the b est
p ossible in the following sense There are mo dule descriptions for which no b end
grid layout exists even if rotation and ipping is allowed
e
e
v for an edge e which connects vertices u and v and u v for We use the notation u
a directed edge e from u to v
Planarity Checking
The planar grid layout problem is divided into two indep endent parts The rst part
is checking if the input is planar This is di erent when ippings are allowed or not
Both cases have known linear time solutions The second part is nding a corresp onding
grid layout
In case ippings are not allowed we need to know if the input mo dule description
is planar One may use a linear time algorithm Zak Pin Ami to check if the
mo dule description is planar
In case ippings are allowed we need to check if by ipping some of the mo dules we
can get a planar mo dule description In this case we say that the mo dule description is
ip planar Pinter Pin shows a linear time algorithm to determine if a given mo dule
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
description is ip planar For the drawing algorithm we also need the actual planar mo dule
description which we get by ipping some of the mo dules Following is a description of
Pinter s algorithm
0
Create a new non ordered graph G a mo di cation of the input ordered graph G in
which every vertex of degree three or more is replaced by a wheel as shown in Figure
0
G is called the wheel graph corresp onding to G
The following lemma is a well known corollary of a theorem of Whitney Whi
Figure changing a vertex to a wheel
ìâìâì úîåö éåðéù :4.2 íéùøú
Lemma A planar triconnected graph with no self loops has only one planar repre
sentation up to ipping the whole graph
Proof In Whi it is shown that a planar triconnected graph with no self lo ops has the
same faces in any planar drawing The meaning of the same face there is the same set
of edges b elonging to the face Since the graph is nonseparable and has no self lo ops a
vertex may not app ear more than once in a face Every face is a simple cycle Therefore
there are exactly two p ossibilities for the clo ckwise order of the edges in each face when
one is the reverse of the other
We have seen that there are two p ossibilities to order the edges in each face Now
we will lo ok at the relation b etween the selections of the order of each face Let us
lo ok at some planar drawing of the graph on the sphere and at an arbitrary face f of
the drawing Each edge app ears on two di erent faces and when walking clo ckwise along
the two faces we pass the edge in opp osite directions This means that the selection the
order of the face at one side of the edge determines the order of the face on the other
side We lo ok at the order of f in the drawing since the dual graph is connected this
order determines the order of all other faces without having to refer to the drawing The
clo ckwise order of the edges in each face gives us a planar representation We have seen
that the clo ckwise order of the edges in f determines the edge ordering Since there are
two reverse not necessarily di erent p ossibilities for this order there are at most two
planar representations of the graph What is left to show is that the other edge ordering
if di erent is a planar representation and that one of these planar representations is the
ipp ed version of the other This can b e seen by taking a planar drawing on the sphere
and turning the sphere inside out
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Lemma A module description is ip planar if and only if its corresponding wheel
graph is planar
Proof If the wheel graph is planar there is a planar drawing of it Since each wheel is
triconnected it has only one planar representation up to ipping by lemma The order
of the edges around the center of the wheel is either identical to the edge order around
the original vertex or is its reversal By removing the edges and vertices of the rim of the
wheel we get a planar drawing of the original graph
If there is a planar drawing of the graph that corresp onds to a ip planar mo dule
description then by adding a small circle around each vertex in the drawing we get a
planar drawing of the wheel graph Therefore the wheel graph is planar
0
To check if a drawing is p ossible we check if G is planar LEC ET BL HT
0
If G is planar we make it a planar ordered graph CNAO and extract from it an edge
ordering that makes G a planar ordered graph The clo ckwise cyclic order of the edges
incident to a vertex is the order of the corresp onding radial edges in the wheel around the
center This shows what mo dules are ipp ed yielding a planar mo dule description To
summarize
Theorem There is a linear time algorithm that checks if a module description is ip
planar and if it is produces a corresponding planar module description and a speci cation
of the ipped modules
All the algorithms used in this section run in linear time The remaining task is to
construct a planar grid layout for the planar mo dule description at hand
Planar Grid Layouts The Case of No Self
Lo ops
We assume G is connected for otherwise we can draw a planar grid layout of each con
nected comp onent separately The connected comp onents of a graph can b e found in
linear time Thus we show that any graph can b e drawn in linear time by showing
that connected graphs can b e drawn in linear time In this section we assume G has no
self lo ops
We draw G in several stages Following we give an overview of the algorithm and
later discuss the details First some edges may b e added and the edges of the ordered
graph are directed to b ecome an st bip olar orientation Next the rotation of the drawing
of each vertex is determined that is for every vertex we choose one of the four p ossible
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
rotations for its mo dule Following the edges incident to each vertex are divided to edges
that will b e drawn to the right or to the left of the mo dule Then we nd a visibility
representation of the ordered graph with enough space in it to accommo date the drawings
of the mo dules And nally we translate the visibility representation to the grid layout
by drawing the mo dules Now let us present a more detailed description
If the graph is nonseparable we nd an st bip olar orientation of it ET Otherwise
we rst add edges to the ordered graph which make it nonseparable without impairing
its planarity This is done without changing the order around vertices of the edges of the
input ordered graph it can b e done by adding edges inside faces in which a cut vertex
app ears more than once or as describ ed in the previous chapter Next we nd an st
bip olar orientation of the new ordered graph The added edges are called auxiliary and
are not drawn in the grid layout As in ET one may choose any edge to b e s t We
choose an edge such that when it is draw vertically with s at the b ottom the face we
want to b e external is on its left hand side This ensures that the sp eci ed face will b e
external Henceforth we consider the ordered graph to b e directed the edges having the
directions of the st bip olar orientation
Next for each vertex we want to determine the rotation of its mo dule s drawing and
divide its incident edges to two subsets one containing edges that will b e drawn to the
right of the vertex and the other containing edges that will b e drawn to the left of the
vertex
First let us describ e how this is done for vertices which are neither s nor t We need to
know which edges are the leftmost and rightmost edges of the incoming outgoing edges
incident to a vertex Lemma implies that there is no ambiguity in their indication for
vertices other than s and t
Lemma implies that the p erimeter of the mo dule can b e divided to two paths
one incident only to incoming edges and the other incident only to outgoing edges The
mo dule can b e rotated such that the path of outgoing edges has a section on the top of
the mo dule and the path of incoming edges has a section at the b ottom of the mo dule
Figure demonstrates how this can b e done More than one appropriate rotation may
b e p ossible
Figure rotation of a vertex which is not a source nor a sink
øå÷î àìå øåá àì àåäù úîåö ìù áåáéñä úòéá÷ :4.3 íéùøú
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
the marked path on the p erimeter of the mo dule is the path that contains
only outgoing edges and should have a section on the top of the mo dule
etc
We choose a p oint on the path incident to outgoing edges which is on the top of the
mo dule and a p oint on the path incident to incoming edges which is at the b ottom of the
mo dule Such p oints exist due to the way the paths have b een selected These two p oints
divide the p erimeter of the mo dule to two new paths The edges incident to the new path
which contains the right hand side of the mo dule are the edges that will b e drawn to the
right of the vertex The edges incident to the other new path which contains the left hand
side of the mo dule are the edges that will b e drawn to the left of the vertex Although
the auxiliary edges are marked as drawn to the right or to the left of the mo dule they
will not b e actually drawn
Now we deal with s and t First we consider the edge s t to b e the leftmost edge
of s and t Since s t has only outgoing incoming edges we choose a p oint on the
p erimeter of the mo dule b etween the leftmost and rightmost incident edges and consider
it to b e the path incident to incoming outgoing edges The rotation of the mo dule
and the division of the incident edges is done as with the other vertices Note that this
ensures that s t will b e the leftmost edge of the resulting visibility representation and
the correct face will b e external
Next for every vertex we determine how much space the drawing of the vertex needs
We calculate for every vertex a quantity called the drawn height of the vertex The drawn
height is comp osed of the height of the mo dule plus added space b elow and ab ove the
mo dule The size of the added space ab ove b elow the mo dule is the maximum of the
number of edges incident to the top b ottom of the mo dule that are drawn to its left
and the number of edges incident to the top b ottom of the mo dule that are drawn to
its right
Next we want to construct a visibility representation so that each vertex will have
the space needed to draw its mo dule and its connections This requires that if we stretch
each vertex of the visibility representation downwards to b e in the height of its drawn
height the vertices of the visibility representation will remain nonoverlapping We also
require that in the drawing of each vertex v in the visibility representation there will b e
a section having the width of the mo dule of v with the following prop erties All edges
to b e drawn to the right left of v are incident to v at p oints to the right left of the
section therefore no edge is incident to v at this section
Such a visibility representation can b e found using the algorithm of DTT This
algorithm nds a visibility representation for an st bip olar orientation with certain sp ec
i ed directed paths of the st bip olar orientation drawn as straight vertical lines That
is the edges of each such path are vertically aligned The given paths are required not
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
to cross each other in a prop er planar drawing of the ordered graph In this algorithm
the co ordinates of the visibility representation are found by two top ological orderings
A topological ordering for a directed acyclic graph with non negative edge weights is an
assignment of a p ositive integer to each vertex such that for each edge the number as
signed to its destination minus the number assigned to its origin is greater than or equal
to the weight of the edge One ordering is done on the vertices of the original graph
and the other is done on the vertices of a mo di cation of the dual graph The weights
represent the minimal space b elow a vertex in the rst top ological ordering and represent
the minimal space to the right of an edge in the second top ological ordering Following
we describ e how to determine the weights and how to sp ecify the paths to b e aligned
For a vertex that has incoming outgoing edges drawn to the left of the vertex we
will call the rightmost incoming outgoing edge which is drawn to the left of the vertex
aligned at the vertex If a vertex has incoming outgoing edges but they are all drawn to
the right of the vertex we duplicate the leftmost incoming outgoing edge The duplicate
edge will b e considered drawn to the left of the vertex and will b e called aligned at the
vertex The added edge is auxiliary and will not app ear in the grid layout
For every vertex which is neither the source nor the sink of the st bip olar orientation
its incoming aligned edge and outgoing aligned edge will b e put in a path to b e aligned
in the visibility representation This path may continue farther if any of these two edges
are aligned at their other end vertices The paths created do not cross each other in a
prop er planar drawing of the ordered graph since by de nition there can b e at most one
path which passes through a vertex
The weight of an edge in the top ological ordering of the original graph including the
auxiliary edges is the drawn height of its target vertex plus The weight of an edge
in the top ological ordering of the mo di ed dual graph is as follows If the corresp onding
edge is aligned at one of its end vertices its weight is the vertex s mo dule width plus
If the edge is aligned at b oth of its end vertices its weight will have the greater value of
the two mo dule widths plus Otherwise the edge will have a weight equal to
The sp eci cation of the input to the visibility representation algorithm ensures us that
the visibility representation pro duced will b e as required There will b e a section of the
line representing the vertex with length greater than or equal to the mo dule width plus
with no incident edges and it will b e to the right of the edges aligned at the vertex
Finally we translate the visibility representation to the grid layout changing the draw
ing of each vertex We make all the changes to the drawing of the vertex in an area of the
size of a rectangle whose top side is on the line of the vertex in the visibility representa
tion The height of the rectangle is the drawn height of the vertex plus The mo dule
itself is drawn b elow the section of the vertex in the visibility representation that has no
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
incident edges such that it will have the correct amount of space ab ove and b elow it as
calculated in the drawn height See gure
For every edge incident to the right and left sides of the mo dule a horizontal line is
drawn connecting its terminal with the vertical line representing the edge in the visibility
representation For every edge incident to the b ottom and top of the mo dule two lines
are used for the connection A vertical line originating at the terminal and a horizontal
line The horizontal lines of the edges incident to the top b ottom of the mo dule that
are drawn to the right left of the mo dule are stowed next to the mo dule See gure
b
RR R
R L
L
R
a
RR L
c
Figure the steps of drawing a mo dule in the algorithm
íúéøåâìàá ìåãåî ìù øåéöä éáìù :4.4 íéùøú
a after the edges are directed the rotation of the mo dule is determined
and the incident edges divided to edges drawn to the left and to the right
of the mo dule this is done by twice dividing the p erimeter of the mo dule
to two paths b the drawing of the vertex in the visibility representation
c the nal drawing of the mo dule in the grid layout
This pro duces the grid layout The connection of an edge to a mo dule introduces one
b end when incident to the right or left side of the mo dule and two b ends when incident
to the top or b ottom of the mo dule An edge has at most two b ends at the top at most
two b ends at the b ottom and no b ends in b etween Thus an edge has no more than four
b ends in the grid layout
Theorem A linear time algorithm exists that given a planar module description with
no self loops nds a corresponding bend planar grid layout
Planar Grid Layouts The Case With Self Lo ops
Self lo ops imp ose a problem A visibility representation can not represent self lo ops Also
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
some mo dule descriptions with self lo ops have no b end drawings as shown in lemma
Following we describ e how we deal with mo dule descriptions that may have self lo ops
Before drawing a mo dule description we put a new vertex in the middle of every
self lo op that is replace every self lo op by two edges and a new vertex This is shown in
gure b The newly created vertices are called dummy vertices and the vertices of
the original graph are called normal The dummy vertices will b e drawn di erently from
normal vertices they will not b e drawn as mo dules Other than this the graph is drawn
in the same way as describ ed in the previous section
After introducing the dummy vertices the resulting graph is separable and we use the
techniques describ ed in the previous section to make it nonseparable For each normal
vertex we determine the rotation of the mo dule and the placement to the right or to the
left of the incident edges as in section
Next we nd a visibility representation of the graph with enough space in it to accom
mo date the drawings of the vertices The weights of the edges in the top ological ordering
of the original graph are determined as b efore when the drawn height of dummy vertices
is zero The weights of the edges in the top ological ordering of the mo di ed dual graph
are determined as b efore when no edge is considered aligned at a dummy vertex
When translating the visibility representation we do not change the drawing of the
dummy vertices we just change the drawing of the normal vertices as b efore The hor
izontal line representing a dummy vertex together with the drawing of the two edges
incident to it constitute the drawing of the self lo op in the grid layout
t w t
t v
w
u
v
s u
v
L RR
v
v
v u
u
s u
w
u
a b c d e f
L RR
Figure drawing a graph that has self lo ops
íééîöò íéâåç ìòá óøâ øåéö :4.5 íéùøú
the steps of nding the drawing are shown a the input b the input
graph after changing the self lo ops the choice of s t and of the external
face c the st bip olar orientation the edges are drawn upward according
to their direction d the rotation of the mo dules and the division of edges
to b e drawn to the left or to the right of the mo dule e the visibility
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
representation f the nal grid layout
This yields a grid layout of a mo dule description with self lo ops A demonstration run
of the algorithm is describ ed in gure Edges which are not self lo ops have as b efore
at most four b ends Self lo ops are comp osed of two edges and a dummy vertex in the
visibility representation Each of these two edges has at most two b ends near the normal
vertex the self lo op is incident to and exactly two b ends at the drawing of the dummy
vertex in the visibility representation Thus self lo ops have a total of at most six b ends
Theorem A linear time algorithm exists that given a planar module description
nds a corresponding bend planar grid layout in which every edge that is not a self loop
has at most bends
The co ordinates of the drawing are determined by the two top ological orderings there
fore the maximal co ordinate is smaller than the sum of the edge weights We denote the
sum of the widths of the mo dules in their rotation in the grid layout by W and the sum
of the heights of the mo dules by H So the width and height of the pro duced drawing are
W O jE j H O jE j resp ectively
Mo dule Descriptions that are Hard to Draw
There are linear time algorithms to check if a mo dule description is planar or ip planar
We have shown a linear time algorithm that given a planar mo dule description nds a
corresp onding b end planar grid layout in which every edge that is not a self lo op has at
most b ends Thus the whole pro cess of nding a planar grid layout for a given mo dule
description if one exists can b e done in linear time As far as the upp er b ounds on the
number of b ends in each edge are concerned we give examples of planar mo dule descrip
tions which demonstrate that our b ounds are tight See gure These examples do
not dep end on mo dule ippings or on the choice of the external face
a b
Figure planar mo dule descriptions that do not draw well
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
áèéä íéøééèöî àìù íééøåùéî íéìåãåî éèì÷ :4.6 íéùøú
a requires at least b ends for some edge b requires at least b ends for
some self lo op
Lemma The module description in gure a has no bend grid layout
Proof We show that any grid layout of the mo dule description has an edge that must
have at least b ends Notice that ipping only one of the mo dules would make the
mo dule description non planar First we lo ok at the rotation of the two mo dules only
their relative angle is imp ortant For any planar grid layout there is a side A of the rst
mo dule and a side B of the second mo dule such that
A faces the same direction as the opp osite side of B
an edge connects A and B
an edge connects the opp osite sides of A and B
One of these edges needs at least b ends
Lemma The module description in gure b has no bend grid layout
Proof Each of the self lo ops needs at least b end to b e drawn if the face comp osed only
of that self lo op is not the external face Since only one of these faces can b e external at
least one of the self lo ops has at least b ends
An Application
The drawing algorithm we presented can draw the grid layout with any sp eci ed external
face This is imp ortant since it enables us to pro duce a planar grid layout including
external connections
This is demonstrated in the following example which is an application of the drawing
algorithm
We consider constructing a planar grid layout intended to b e the internal layout of
a big mo dule Ideally we would get the description of the big mo dule consisting of its
size the placement of its terminals and the mo dule description of its inside That is
in addition to the inner mo dule description its external face is sp eci ed as well as the
following constraints on external terminals For each external terminal a connection to
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
a terminal on the p erimeter of an inner mo dule is sp eci ed Notice that the additional
sp eci cation of the terminals of the big mo dule may cause the input to b ecome nonplanar
Unfortunately the problem of nding if such a planar grid layout exists is hard Consider
an input with no edges The problem b ecomes the D bin packing problem which is NP
hard In certain cases the following approach solves the problem Find a grid layout of
the inside of the mo dule and if it is not to o big connect it to the p erimeter of the big
mo dule This can b e done as follows
Add four sp ecial vertices to the inner mo dule description These vertices represent the
sides of the big mo dule s p erimeter The sp ecial mo dules are connected to the original
mo dule description by edges representing the terminals of the big mo dule and the sp ecial
mo dules are also connected to each other see gure a Also the sp ecial mo dules
will b e drawn in the same rotation as they have in gure a This is p ossible since
these rotations comply with the rules of selecting a rotation and one may select any such
compliant rotation A grid layout of the new mo dule description is constructed using our
algorithm Also the s t is chosen as in gure a
the st edge
a b
the
original
c
mo dule
description
Figure an example of an application planar layout of a big mo dule
ìåãâ ìåãåî ìù úéøåùéî äùéøô - íåùééì àîâåã :4.7 íéùøú
a the new mo dule description b the grid layout pro duced after removing
edges b etween the sp ecial mo dules c the nal layout of the big mo dule
If the grid layout pro duced is bigger than the size given for the big mo dule we fail
If not erase from the grid layout the edges that connect the sp ecial mo dules see g
ure b Remove the drawings of the sp ecial mo dules and change the drawing of the
remaining edges connected to them so they connect to the terminals of the big mo dule
on its p erimeter This is demonstrated in gure c
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Chapter
Non Planar Drawings
In this chapter we consider the non planar versions of the problems we dealt with b efore
Namely there is no requirement that that the input or the output b e planar We allow
edges to cross one another in the drawings In this chapter the fact that a drawing is
called k b end do es not imply that the drawing is planar
Vertices as Points
We call a graph with no self lo ops which has no vertex of degree greater than d a d
graph Biedl Bie describ es a linear time algorithm that nds a b end drawing for
every graph
Techniques similar to Biedl s can b e used to nd in linear time a b end drawing for
any simple Graph
It is trivial to nd a graph with no b end drawing and a graph with no b end
drawing
Vertices as Mo dules Grid Layouts
In this section we consider the nonplanar versions of the problems we dealt with in chap
ter The input mo dule description is not assumed to b e planar and even if it is the
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
output grid layout is not required to b e planar
We show how to nd b end nonplanar grid layout of any mo dule description The
ippings of the mo dules and the angle in which they are drawn may b e determined
arbitrarily The input to the algorithm is a mo dule description and it includes the rotation
in which each mo dule is to b e drawn A linear time drawing algorithm is describ ed in
gure The algorithm is demonstrated in gure The algorithm works by rst
placing the mo dules diagonally so that if we continue the line of any terminal it do es
not hit any mo dule Next we complete the drawing of the edges
pro cedure NPGL M
f M is a module description with a rotation speci ed for each module g
b egin
place the mo dules in the drawings diagonally such that
the drawings of two mo dules do not share any row or column
R the smallest rectangle on the grid which contains all mo dules
for every edge e of M do
b egin
continue the lines of the two terminals of e till they exit from R
and connect them by at most lines around the b oundary of R
expand R to contain the drawing of e
end
end
Figure construction of a b end nonplanar grid layout
(úéøåùéî àì) úéúééðô-4 úéâéøñ äùéøô úééðá :5.1 íéùøú
Figure output of the nonplanar grid layout algorithm
úéøåùéî àìä úéâéøñä äùéøôä úàéöî íúéøåâìà ìù èìô :5.2 íéùøú
the edges are numbered in the order in which they are added by the algo rithm
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Theorem Algorithm NPGL see gure nds in linear time a bend nonplanar
grid layout for any module description with any speci ed rotations of the modules
The area of the grid layouts has the same upp er b ound as for the planar grid layouts
Finally we mention two mo dule descriptions which have no b end grid layout The
rst mo dule description consists of one mo dule which has one self lo op The terminals of
the self lo op are on opp osite sides of this mo dule Clearly the self lo op needs at least
b ends The second mo dule description has two mo dules and sixteen edges Every side of
one mo dule is connected by one edge to every side of the other mo dule
Lemma A module description that has two modules and sixteen edges such that
every side of one module is connected by one edge to every side of the other module has
no bend grid layout
Proof We lo ok at any grid layout of the mo dule description and show an edge that must
have at least b ends in this grid layout First we lo ok at the placement and the rotation
of the two mo dules The mo dules are b oth drawn non overlapping on the grid therefore
the following holds There is a side A of the rst mo dule and a side B of the second
mo dule such that A and B are parallel and the terminals exit A and B in opp osite
directions Every side of the rst mo dule is connected by one edge to every side of the
second mo dule Therefore one edge connects A and B and it must have b ends in the grid layout
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
Biblio graphy
Ami A Amiho o d A direct linear time planarity test for un ippable mo dules In
tern J Computer Math vol pp
Bie T Biedl Embedding nonplanar graphs in the rectangular grid Technical
Rep ort RUTCOR Rutgers University
BK T Biedl and G Kant A b etter heuristic for orthogonal graph drawings In
ESA to appear in Lecture Notes in Computer Science Springer Verlag
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 Journal Comput
System Sci vol pp
CNAO N Chiba T Nishizeki S Ab e and T Ozawa A linear algorithm for embed
ding planar graphs using PQ trees Journal Comput System Sci vol pp
DETT G Di Battista P Eades R Tamassia and I G Tollis Algorithms
for drawing graphs an annotated bibliography This do cument is avail
able via anonymous ftp from wilma cs brown edu les
pub papers compgeo gdbiblio tex Z and
pub papers compgeo gdbiblio ps Z
DTT G Di Battista R Tamassia and I G Tollis Constrained visibility represen
tations of graphs Information Processing Letters vol pp
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
ET S Even and R E Tarjan Computing an st numbering Theoretical Computer
Science vol pp
Eve S Even Graph Algorithms Computer Science Press
HT J Hop croft and R Tarjan E cient planarity testing J Assoc Comput
Mach vol pp
Kan G Kant Drawing planar graphs using the lmc ordering In Proc of the rd
Annual IEEE Conference on Foundations of Computer Science pp
LEC A S Lemp el S Even and I Cederbaum An algorithm for planarity testing of
graphs In P Rosenstiehl editor Theory of Graphs International Symposium
Rome pp New York Gordon and Breach
MO Z Miller and J B Orlin NP completeness for minimizing maximum edge
length in grid embeddings Journal of Algorithms vol pp
Pin R Y Pinter River routing Metho dology and analysis In third CALTECH
conference on Very Large Scale Integration pp Computer Science
Press
RT P Rosenstiehl and R E Tarjan Rectilinear planar layouts and bip olar ori
entations of planar graphs Discrete and Computational Geometry vol no
pp
Shi Y Shiloach Linear and Planar Arrangements of Graphs PhD thesis Depart
ment of Applied Mathematics Weizmann Institute of Science Rehovot Israel
Sto J A Storer On minimal node cost planar embeddings Networks vol
pp
Tam R Tamassia On embedding a graph in the grid with the minimum number
of b ends SIAM J Computing vol no pp
TDB R Tamassia G Di Battista and C Batini Automatic graph drawing and
readability of diagrams IEEE Transactions on Systems Man and Cybernet
ics vol no pp
TT R Tamassia and I G Tollis A uni ed approach to visibility representations
of planar graphs Discrete and Computational Geometry vol no pp
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994
TT R Tamassia and I G Tollis Planar grid embedding in linear time IEEE
Transactions on Circuits and Systems vol no pp
TTV R Tamassia I G Tollis and J S Vitter Lower b ounds for planar orthogonal
drawings of graphs Information Processing Letters vol pp
Whi H Whitney Congruent graphs and the connectivity of graphs American
Journal of Mathematics vol pp
Zak S Zaks An easy planarity test for un ippable mo dules Technical Rep ort
Computer Science Department Technion Israel Inst of Tech
Technion - Computer Science Department M.Sc. Thesis MSC-1994-01 1994