An Algorithm for Drawing Planar Graphs
Bor Plestenjak
IMFM TCS University of Ljubljana Jadranska SI Ljubljana Slovenia
email b or plestenjak fmf uni lj si
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
SUMMARY
A simple algorithm for drawing connected planar graphs is presented It is derived from the
Fruchterman and Reingold spring emb edding algorithm by deleting all repulsive forces and xing
vertices of an outer face The algorithm is implemented in the system for manipulating discrete
mathematical structures Vega Some examples of derived gures are given
Key words Planar graph drawing Force directed placement
INTRODUCTION
A graph G is planar if and only if it is p ossible to draw it in a plane without any edge intersections Every
connected planar graph admits a convex drawing i e a planar drawing where every face is drawn as
a convex p olygon We present a simple and e cient algorithm for convex drawing of a connected planar
graph In addition to a graph most existing algorithms for planar drawing need as an input all the faces of
a graph while our algorithm needs only one face of a graph to draw it planarly
Let G b e a connected planar graph with a set of vertices V fv v g and a set of edges E
1 n
f u v u v g Let W fw w w g V b e an ordered list of vertices of an arbitrary face in
1 1 m m 1 2 k
graph G which is chosen for the outer face in the drawing
The basic idea is as follows We consider graph G as a mechanical mo del by replacing its vertices with
metal rings and its edges with elastic bands of zero length Vertices of the outer face W are xed in a vertices
of a regular p olygon of a size k while all other vertices are let free Under the in uence of attractive forces in
bands free vertices will move until the system nally reaches an equilibrium The layout in the equilibrium
is a convex drawing
The algorithm is included in package Vega under the name SchlegelDiagram Vega is Mathematica
based system for manipulating graphs and other combinatorial structures such as groups maps etc devel
op ed by IMFM TCS Ljubljana Vega is mainly written in Mathematica however it also includes external
programs for time complex op erations e g the implementation of our algorithm
ALGORITHM
The algorithm is based on the force directed placement graph drawing algorithm by Fruchterman and Rein
gold In each step the algorithm calculates the e ect of attractive forces on each vertex and then moves
the vertex in the direction of the resultant force The displacement in step i is limited to some maximum
value cool i that we call temp erature and which satis es lim cool i As the temp erature decreases
i!1
to zero smaller and smaller displacements are allowed and after some numb er of steps displacements are so
small that we can say that the layout freezes The algorithm ends after the prescrib ed numb er of iterations
and if the forces and the co oling function cool i are chosen prop erly the nal layout is so near to the
equilibrium that it is planar
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
For the force b etween two adjacent vertices u and v we use the third order law
3
F C d
uv
where C is a constant and d ku pos v posk is the distance We tested di erent order p owers and the
result is a compromise b etween go o d results and an e cient calculation
The algorithm consists of the following steps
Position all vertices of an outer face W in vertices of a regular p olygon of size k inscrib ed into the unit
circle and put all other vertices in the origin
For i to iter ations
a For all vertices v V set the resultant force F to zero
v
F
v
b For all edges u v E calculate the attractive force F and up date the resultant forces F and
uv u
F
v
3
F C v pos u pos
uv
F F F
u u uv
F F F
v v uv
c For all vertices v V W move vertex v in the direction of the resultant force F for the size
v
of the force but not more than for the value of cool i
F
v
v pos v pos min jF j cool i
v
jF j
v
Constant C in has to b e in harmony with the maximum displacement Namely if C is to o large
then in every vertex is shifted for the maximum displacement irresp ective of the force size On the other
hand if C is to o small then the displacements are small and the algorithm do es not reach the equilibrium
in a prescrib ed numb er of steps
If the temp erature decreases to o slowly then displacements are to o large and to o many steps are needed
to reach the equilibrium Yet if the temp erature decreases to o fast then the layout freezes in a non planar
drawing
A suggested values for constant C and function cool i are
r
n
C
p
n
cool i
3 2
i
n
p
The expression is the average area b elonging to one vertex and therefore we take for the average
n n
in and distance b etween adjacent vertices This justi es the presence of term n
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
Figures to demonstrate how the algorithm works The graph is the icosahedral fullerene isomer of
C derived in Vega from the do decahedron using two leapfrog transformations We usually pick the
180
largest face for the outer face but since p entagons are arranged in a shap e of a do decahedron we cho ose a
p entagon in order to exhibit symmetry For C and cool i we apply and resp ectively It can b e seen
from these gures how the edges closer to the b order untwist rst and how they force those in the middle to
unfold until nally the layout b ecomes planar
Figure C after iterations Figure C after iterations Figure C after iterations
180 180 180
MODIFICATIONS
Periphericity If the algorithm is applied to a graph with a large numb er of faces e g a fullerene on
many vertices then it pro duces a drawing with large faces on the b order and large numb er of crowded small
faces in the middle of a gure
0 0
1 1
2 2
2 1
3 2
2 2
1 1
0 0
Figure Periphericity of a vertex
In order for the algorithm to terminate with approximately equally arranged faces we assume that the
co e cient C in is not equal for all bands Bands at the b order should b e stronger and bands in the
middle should b e weaker For this purp ose we intro duce a p eriphericity of a vertex as the path distance from
the vertex to the outer cycle The de nition of the p eriphericity is more evident in Figure where vertices
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
of a graph are lab eled according to their p eriphericities
The co e cient of the band b etween vertices u and v is a function of p eriphericities u per and v per
and we denote it by C u per v per Let maxper b e the maximum p eriphericity in the graph As we go
through all edges u v E the bands with u per v per should b e the strongest and the bands
with u per v per maxper should b e the weakest Best results are obtained when the co e cients
C u per v per form a geometric progression in the form
r
n maxper u per v per
exp A C u per v per
maxper
For the constant A we prop ose A but other values can b e used as well pro ducing di erent planar
layouts
Figure C with A after Figure C with A after Figure C with A after
540 540 540
iterations iterations iterations
Figures to demonstrate the e ect of di erent values of A The graph is the icosahedral fullerene
isomer of C In Figure A which equals the original algorithm In Figure A and faces in
540
the middle are more equally arranged The value A in Figure is to o large and the obtained drawing
has large faces in the middle that squeeze the rest of the faces In all three case the co oling function is
applied
Oscil lation After some initial numb er of steps the vertices b egin to oscillate b etween two states If
we compare the co ordinates with the co ordinates from two steps ago we can exit the algorithm when the
di erence for every vertex is b elow the prescrib ed constant This approach reduces the numb er of steps
without doing any real harm to the layout quality For the example we take the graph C and
180
3 4 5
with A For and the algorithm ends after and steps resp ectively
Di erences b etween layouts are not visible with the naked eye For instance the maximum di erence b etween
3
the co ordinates of the same vertex after and steps is
Non planar graphs Although the algorithm is derived for connected planar graphs it can b e applied
to other graphs as well For every graph we can x an arbitrary set of vertices on the regular p olygon apply
a mechanical mo del to a graph and search for an equilibrium Although the nal layout do es not need to b e
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
planar it usually contains useful information ab out the symmetry and other prop erties of the graph
EXAMPLES
The following examples are all obtained using Vega We use Vega to determine faces of a planar graph using
a PQ tree algorithm for planarity testing and emb edding in linear time implemented by J Marin cek
Each face of a planar graph is a go o d choice for the outer face but usually we get nicer drawings if we cho ose
the largest face It is advisable to compare drawings for di erent choices of the outer face and then cho ose
5
the most suitable one For all examples in this section we use with A and When
present the term it in gure captions denotes the iteration in which the algorithm reached and exited
Figure and Figure are drawings of the Herschel graph The Herschel graph is a well known
example of a planar nonhamiltonian p olyhedron Notice how the di erent choice of the outer face a ects
the layout
Figure Herschel graph Figure Herschel graph
n m it n m it
Next two gures represent the Tutte graph and the Grinb erg graph These two graphs are well known
examples of planar cubic connected nonhamiltonian graphs Figure is the gure of the Tutte graph
using the largest face for the outer face but nicer and more familiar gure is Figure where the outer face
has size Figure represents the Grinb erg graph
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
Figure Tutte graph Figure Tutte graph Figure Grinberg graph
n m it n m it n m it
Next example presents a planar graph which is not connected Figure is the drawing determined
automatically by the algorithm while Figure is the gure calculated after we manually add two temp orary
edges that are colored gray When we delete temp orary edges we obtain Figure By comparing Figure
and Figure it is easy to see that some edges are overlapping in Figure
This trick can b e applied to every planar graph which is not connected First we add so many temp orary
edges that the graph b ecomes connected and then we use the algorithm which returns the convex drawing
Since the planar layout remains planar if we delete an arbitrary numb er of edges we simply delete all the
temp orary edges and end with a planar drawing of the original graph
The problem is how to add the temp orary edges automatically One of the solutions is to use the
dimensional sub division which adds a new vertex in the center of each face and joins it by edges with
all vertices of the face This transformation is implemented in Vega
Figure Graph with overlapping Figure Graph in Figure with Figure Graph in Figure with
edges added temporary edges out overlapping edges
Planar layouts of p olyhedra are often more surveyable as their three dimensional layouts An example
is the snub do decahedron depicted in a planar and in a three dimensional layout in Figure and Figure
resp ectively The graph in Figure is the medial Grinb erg graph derived from Grinb erg graph using
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
transformations on maps implemented in Vega
Figure Planar layout of snub Figure D layout of snub Figure Medial Grinberg graph
dodecahedron it dodecahedron it
Figure displays the result of applying the algorithm to a well known non planar Petersen graph and
cho osing any cycle of length for the outer cycle Figures and are the results of the algorithm used on a
triangulation of a triangle In Figure a regular p olygon is chosen for the outer face and the gure do es not
have a triangular shap e The choice of three vertices of valence two in Figure pro duces a non triangular
shap e again This problem can b e solved by allowing the user to x a set of vertices in an arbitrary two or
even three dimensional con guration
Figure Petersen graph Figure Triangulated triangle Figure Triangulated triangle
it
CONCLUSIONS
Planar layouts of planar graphs which are often more surveyable as corresp onding three dimensional layouts
can b e easily pro duced using this algorithm This particularly holds for p olyhedral graphs for instance
fullerenes
One of the b ene ts of the algorithm is its time complexity One step of the algorithm has time complexity
O jE j jV j while for instance one step of the algorithm by Fruchterman and Reingold has time complexity
AN ALGORITHM FOR DRAWING PLANAR GRAPHS
2
O jE j jV j A weakness of the algorithm is that it can b e used only for connected planar graphs This
means that planar graphs like trees can not b e drawn automatically using this algorithm
The algorithm o ers many improvements One of them is to allow the user to x a set of vertices in an
arbitrary con guration The algorithm can also b e easily generalized to D layouts
Acknowledgement The author would like to thank Prof T Pisanski at IMFM TCS Ljubljana for his
supp ort and many useful suggestions
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