An Evolutionary Algorithm for Graph Planarisation by Vertex Deletion
Rodrigo Lankaites Pinheiro
1
, Ademir Aparecido Constantino
2
, Candido F. X. de Mendonc¸a
3
and Dario Landa-Silva
1
1
School of Computer Science, University of Nottingham, NG8 1BB, Nottingham, U.K.
2
Informatics Department, State University of Maring´a, Maring´a, PR, Brazil
3
School of Arts, Science and Humanities, USP-East, S˜ao Paulo, SP, Brazil
Keywords:
Graph Planarisation, Evolutionary Algorithms, Vertex Deletion.
Abstract:
A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic
algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The
proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion
number of a graph G = (V, E) is the smallest integer k ≥ 0 such that there is an induced planar subgraph of
G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NP-
complete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work
proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This
constructive heuristic has time complexity of O(n + m), where m = |V| and n = |E|, and it is based on the
PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means
of case studies.
1 INTRODUCTION
Practical applications of Graph Drawing, such as the
design of VLSI circuits, requires drawing techniques
for non-planar graphs. A graph (representing the cir-
cuit) needs to be drawn on the plane (an electronic
chip) without crossing edges. However, graph draw-
ing algorithms are restrained to planar graphs, oth-
erwise the results obtained by these algorithms are
compromised. Network design and analysis and com-
putational geometry are additional well known fields
where the drawing of planar graphs are required. A
possible way to tackle non-planarity in graphs is to
consider its topologicalinvariants, such as the number
of vertex deletion, which can be used as the measure
of non-planarity.
The simple drawing of a graph G = (V, E) is a
drawing of G on the plane, where each edge does not
cross itself, adjacent edges do not cross themselves,
the crossing of two edges only occurs once, the edges
do not cross over vertices, and no morethan two edges
cross at the same point. A graph is considered planar
when there is a simple drawing for this graph on the
plane, without crossing edges. Without loss of gen-
erality, from now on we are considering only simple
drawings.
Take all drawings of G, the drawing which pos-
sesses the lowest number of edge crossings among all
drawings is named optimal drawing of G. And the
number of edge crossings is named crossing number
of G, denoted by cr(G).
The number of vertex deletion Φ(G) is the small-
est integer k > 0 such that the deletion of k vertices
from G produces a planar graph. The decision prob-
lem regarding the number of vertex deletion, the num-
ber of vertex splitting, the number of edges deletion
and the crossing number are all NP-complete (Faria
et al., 2001a; Garey and Johnson, 1983; Liu and Geld-
macher, 1977; Yannakakis, 1978). (Faria et al., 2006)
proved that an approximation algorithm cannot exist
for the graph planarisation problem using the vertex
deletion operation, hence a heuristic algorithm be-
comes a viable alternative to tackle the problem. Also
it has been shown that it remains NP-hard even for
cubic graphs (Faria et al., 2001a; Faria et al., 2001b;
Faria et al., 2004). Besides, (de Figueiredo et al.,
1999) showed that the same occurs for the number
of vertices splittings according to the result obtained
by (Robertson and Seymour, 1995).
Literature reports many algorithms that attempt to
remove a minimal number of edges to obtain a planar
subgraph (Chiba et al., 1979; Fisher and Wing, 1966;
464
Lankaites Pinheiro R., Constantino A., F. X. de Mendonça C. and Landa-Silva D..
An Evolutionary Algorithm for Graph Planarisation by Vertex Deletion.
DOI: 10.5220/0004883704640471
In Proceedings of the 16th International Conference on Enterprise Information Systems (ICEIS-2014), pages 464-471
ISBN: 978-989-758-027-7
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
Marek-Sadowska, 1978; Ozawa and Takahashi, 1981;
Pasedach, 1976). One of the best approaches is the
PLANARISE algorithm by (Jayakumar et al., 1989)
referred as JTS PLANARISE algorithm. The JTS PLA-
NARISE algorithm is based on the planarity test al-
gorithm by (Lempel et al., 1967) and (Even, 2011)
(also referred as the LEC algorithm) and its imple-
mentation using PQ-trees (Booth and Lueker, 1976).
(Eades and de Mendonc¸a, 1993) considered the num-
ber of vertex splittings adapting the PLANARISE al-
gorithm into the SPLIT-PLANARISE. That was done
by replacing the edge removaloperation for the vertex
splitting operation. Both algorithms have time com-
plexity O(n
2
) and space complexity O(n+ m), where
n represents the number of vertices and m represents
the number of edges of G. This work proposes an al-
gorithm named VD-PLANARISE which uses similar
ideas to the JTS PLANARISE algorithm above men-
tioned, though it uses the operation of vertex deletion
instead of edge removal. In the next section it will
be discussed how the JTS PLANARISE algorithm was
adapted for the new proposed constructive heuristic
which has time and space complexity of O(n + m).
We can also highlight that the JTS PLANARISE algo-
rithm generates a planar subgraph where the proposed
algorithm generates an induced planar subgraph.
Section 2 describes a few necessary concepts. In
section 3 we present the VD-PLANARISE algorithm.
The section 4 analyses the complexity and the perfor-
mance of the proposed algorithm. Section 5 presents
the evolutionary algorithm MAVD-PLANARISE and
later we show an empirical analysis of the performed
tests of the proposed algorithms (section 6).
2 THE LEC ALGORITHM AND
PQ-TREES
This section presents the basics of the JTS PLA-
NARISE algorithm, which is based on the LEC pla-
narity test algorithm which is performed with the aid
of PQ-trees. The definitions of the data structure and
its operations are described in this section. However,
for further details on the implementation of the opera-
tions on PQ-trees we recommend the work of (Booth
and Lueker, 1976).
The LEC algorithm only deals with biconnected
graphs. Considering that it is fairly easy to divide a
graph into a tree of biconnected components (blocks),
(Gibbons, 1985) presents a linear complexity algo-
rithm for the generation of a tree of biconnected com-
ponents for a given graph. This work may consider,
thus, only biconnected graphs.
Take a biconnected graph G = (V, E) with n = |V|
vertices and m = |E| edges. An st-numbering is a
labeling of the vertices in G with integer numbers
1, 2, , n where 1 is adjacent to n and a vertex num-
bered j is adjacent to a pair of vertices numbered i
and k where i < j < k. The vertex 1 is named source
and is referred as s while the vertex n is named sink
and is referred as t. Each biconnected graph has a
st-numbering (Lempel et al., 1967) and such labeling
can be found in linear time (Even and Tarjan, 1976).
The graph G labeled with st-numbers is named st-
graph.
Let G
k
, where 1 ≤ k ≤ n, be a subgraph of an st-
graph G induced by the set of vertices V
k
= 1, 2, ..., k.
Let B
k
be a graph associated with the subgraph G
k
and
all of the edges of G connected with the vertices V
k
and V −V
k
in G. These edges are named virtual edges
and the verticesV −V
k
are named virtual vertices. The
virtual vertices are labeled as its original vertices in G;
though they remain apart (a leaf for each adjacent ver-
tex not yet embedded). Consequently, in B
k
there may
be several virtual vertices with the same label, each of
them with exactly one virtual edge. A drawing B
k
is
named bush form of G
k
if the vertices with smaller or
equal labels than k appears at a higher level than the
leaves and all of the virtual vertices appears as leaves.
It is possible to demonstrate (Even, 2011; Lempel
et al., 1967) that a st-graph is planar if and only if
for each B
k
, 2 ≤ k ≤ n− 2, there is a planar graph B
′
k
isomorph to B
k
such that all the virtual vertices in B
′
k
labeled k+ 1 appear consecutively.
A PQ-tree (Booth and Lueker, 1976) T is a data
structure that represents a set of permutations in a set
S. The nodes of T can be leaves, representing the
elements of S; P-nodes, conventionally represented as
a circle; and Q-nodes, conventionally represented as
a rectangle.
For this kind of tree the order that the descendants
of a node appear is important. The borderline of T
is defined as the permutation represented by the order
of the leaves of T from left to right. For example, the
borderline of the first PQ-tree in Figure 1 is [abcde].
The set of permutations represented by T is gen-
erated by rearranging the descendants of each node P
and Q, according to two rules – the descendants of a
P-node can be freely permuted and the order of the
descendants of a Q-node can only be inverted.
The set of permutations of S represented by T is
the set of borderlines of the PQ-trees obtained from
T, by rearranging the descendants according to these
rules. For example, the set of permutations repre-
sented by the PQ-tree in Figure 1 is: [abcde], [abced],
[cbade], [cbaed], [dabce], [dcbae], [eabcd], [ecbad],
[deabc], [decba], [edabc], and [edcba].
These PQ-trees proved to be useful in many prob-
AnEvolutionaryAlgorithmforGraphPlanarisationbyVertexDeletion
465
Figure 1: The twelve permutations allowed for the given PQ-tree.
lems involving a successive reduction of the set of
permutations to find a specific permutation. For ex-
ample, they have been used to identify planar graphs,
interval graphs, matrix with the property of consecu-
tive ones (Booth and Lueker, 1976), hierarchical pla-
nar graphs (Battista and Nardelli, 1988), as well as
the dominance drawings (Eades and de Mendonc¸a,
1993).
In this work, a PQ-tree T
k
is used to represent the
bush form B
k
in the algorithm. The nodes of T
k
corre-
spond to the following:
• leaves: the virtual vertices of B
k
;
• Q-nodes: the maximal biconnected components
in B
k
; and
• P-nodes: the articulation vertices in B
k
.
The leaves are named pertinent if they correspond
to the next selected vertices (label k+ 1) with the pos-
sibility to be embedded, while the others are named
non-pertinent leaves. In the same way, a non-leaf
node X is pertinent if any leaf descendant of X in the
PQ-tree is pertinent. If all the leaves from the descen-
dants of a node X in the PQ-tree are pertinent, then
X is named a full node. If no leaf descendant of the
node X is pertinent then X is empty. The X border-
line is defined by its set of descendant leaves, read
from left to right. A node X is a pertinent root if it is
the lowest level node whose borderline has only per-
tinent leaves. The tree rooted in X is named pertinent
subtree. Once a pertinent root is identified, a series of
pattern tests and reallocations described in (Booth and
Lueker, 1976) can be used in order to build a new tree
in which all the pertinent leaves are shown consecu-
tively if such tree exists. In this case all the pertinent
leaves in the new tree will appear as descendants of
a single node. For instance, suppose that the PQ-tree
from Figure 2 represents a bush form B
7
; P1 node is
a pertinent node; Q1 node is an empty node; P2 node
is the pertinent root; Q2 node is a full node; the per-
tinent leaves are labeled as 8; and the non-pertinent
leaves are labelled as 9,10,11,12.
Figure 2: A PQ-tree of a bush form B
7
.
Reduction is an important operation of a PQ-tree.
On an abstract level, the reduction takes a set of per-
mutations Π of S and a subset S
′
⊆ S and returns a
subset Π
′
of Π in a way that the elements of S
′
con-
secutively appear in all the permutations in Π
′
. The
elements of S
′
are named pertinent elements of S.
(Booth and Lueker, 1976) created an algorithm
that reduces a T
k
tree into a T
∗
k
tree in a way that all the
pertinent leaves consecutively appear in the border-
line (when possible). The reduction operation can be
efficiently executed with a sophisticated implementa-
tion of PQ-trees. This work, however, does not dis-
cuss these operations that are detailed in (Booth and
Lueker, 1976).
It is trivial to notice that not always a PQ-tree T
k
can be reduced into a T
∗
k
, thus (Ozawa and Takahashi,
1981) defined some criteria to test a tree before ap-
plying the reduction. Let G be a biconnected st-graph
and T
1
,T
2
,,T
n−1
be the PQ-trees corresponding to the
bush forms B
1
,B
2
,,B
n−1
of G. A node X of a PQ-tree
is classified according to its borderline, as follow:
• Type A: if the rooted subtree in X could be rear-
ranged in a way that all the pertinent leaves de-
ICEIS2014-16thInternationalConferenceonEnterpriseInformationSystems
466
scendant of X consecutively appear in the middle
of the borderline, with at least a non-pertinent tree
in each extreme of the borderline. For example,
the P
1
node in Figure 2 is the type A.
• Type B: if the borderline of the rooted subtree in
X consists only of pertinent nodes, then X is a full
node. For example, Q
2
node in Figure 2 is type B.
• Type H: if the rooted subtree in X could be rear-
ranged in a way that all the pertinent leaves de-
scendant of X consecutively appear in one of the
ends of the borderline. For example, P
2
node in
Figure 2 is type H.
• Type W: if the borderline of the rooted subtrees in
X consists only of non-pertinent leaves, that is, X
is an empty node. For example, Q
1
node in Figure
2 is type W.
It is known that a PQ-tree is not always one of the
types A, B, H or W. However, the need to transform
(essentially by vertex deletion) the whole tree in a tree
of the W type will be further looked at.
A graph G containing n vertices is planar if
and only if the pertinent roots in all the PQ-trees,
T
2
,T
3
,,T
n−2
of G are of the B, H or A type. A PQ-
tree is reducible if its pertinent root is of B, H or A
type, otherwise, it is irreducible (Ozawa and Taka-
hashi, 1981).
Both T
1
and T
2
trees are reducible. The first one
because it has just a pertinent leaf corresponding to
the edge (v
1
, v
2
), and the second one because it has
only one type of leaf that is the node corresponding to
the virtual vertex n.
3 PLANARISATION BY VERTEX
DELETION
In this section we introduce the proposed graph
planarisation constructive heuristic VD-PLANARISE
which uses the vertex deletion operation.
In general, the VD-PLANARISE algorithm starts
with a vertex and continues with the insertion of one
vertex at a time, building an induced planar subgraph
G
′
of G. The vertices are selected following the label-
ing order introduced by the st-numbering algorithm
(Lempel et al., 1967). Let v be the next candidate ver-
tex to be inserted in the planar subgraph G
′
. Let E
v
be the edge subset incident with the vertex v and the
vertices of G
′
and let
ˆ
E
v
be the subset of other edges
incident with v. For each iteration if a vertex v cannot
be inserted into G
′
then it is removed. The remaining
set of edges (v, u) ∈
ˆ
E
v
is added (as dummy edges) to
the first inserted vertex (vertex 1). This will be done
to each u vertex that does not have another adjacent
with a smaller label comparing to the v label, aiming
to maintain the property of st-numbering.
The proposed algorithm is presented as follows:
VD-PLANARISE
Input: graph G.
Output: an induced planar subgraph of G.
Pre-processing: obtain a valid st-numbering
of G, obtain
small(u)
for every vertex u in
G.
Begin:
build the initial tree
T
1
;
for k:=2 to n-2 do:
{
following the st-numbering
}
if
T
k−1
is reducible then:
reduce;
else
Update(
v
k
);
obtain
T
k
by replacing every
pertinent node from
T
∗
k−1
by
a new P-node
P
k
such as every
edge adjacent to the vertex
v
k
with label higher than k
appears as a direct descendant
of
P
k
.
return G;
End.
The algorithm starts with the T
1
tree and builds the
sequence of PQ-trees T
2
,T
3
,. If a graph is planar the
LEC algorithm finishes after building the T
n−1
tree,
otherwise it finishes when it detects the impossibility
to reduce a T
k
tree into T
∗
k
.
Consider T
k
an irreducible PQ-tree of a non-planar
graph, that is, it is impossible to reduce a T
k
tree into
T
∗
k
. The proposed algorithm adds a new operation
named Update(k). This operation removes all the per-
tinent leaves transforming the T
k
tree in type W. Be-
sides, if any u vertex with a label higher than k + 1,
adjacent to the equivalent vertex of the removed per-
tinent leaves does not have any other adjacent vertex
with a smaller label, a new edge (dummy) is added to
the graph in a way that u is adjacent to s. This is nec-
essary to maintain the property of st-numbering. Each
immersion iteration of the algorithm can increase the
number of the adjacent vertices of s. However, this
number does not exceed the number of vertices in G.
The main question is how to inspect the adjacency of
the vertices to be removed in order to assure the prop-
erty of st-numbering without increasing the complex-
ity of time. This can be done by adding a small(u)
field to each vertex u. This field informs the amount
of u adjacents with smaller labels than the u label es-
tablished in the step of st-numbering. Thus, when the
pertinent nodes -correspondent to the v
k+1
vertex - are
removed to make all the T
k
subtrees of the W type, the
small(u) field is reduced by one to each adjacent ver-
tex u of v
k+1
where u has a larger label than k + 1.
AnEvolutionaryAlgorithmforGraphPlanarisationbyVertexDeletion
467
When the value of the small(u) field reaches zero, a
dummy edge is added from s to u. After the last itera-
tion, all the dummy edges from s to u are removed for
vertex where small(u) field is zero.
4 TIME COMPLEXITY AND
PERFORMANCE OF THE
VD-PLANARISE
The Booth and Lueker reduction of all reducible PQ-
trees T
k
can be performed in a total time of O(n+ m)
(Jayakumar et al., 1989). If a PQ-tree T
k
is not re-
ducible, the Update(k) operation that will remove the
pertinent vertices is performed. Suppose the worst
case with the maximum of removed vertices (notice
that it is true for the K
n
graph). In this case, for each
v vertex removed the algorithm inspects the labels of
each adjacent vertex u of v. If the label of u is larger
than the label of v, a unit is reduced to the small(u)
value. Thus the Update(k) operation will inspect each
vertex and its adjacents (like BFS algorithm). Hence
in the worst case the total time of this operation is
O(n+ m). The addition of dummy edges to the s ver-
tex is done in the worst case n times. Therefore the
complexity of time of VD-PLANARISE is O(n+ m).
Since the proposed algorithm is a heuristic one,
questions may arise regarding the quality of its solu-
tions, i.e. the amount of vertices removed. The algo-
rithm efficiency - with the exception of a few cases
such as the complete graph K
n
- is highly dependable
on the st-numbering, since the PQ-trees are built in
that order. Hence remains the question: how many
different st-numberings can a graph possess? And
what is the impact of different st-numberings regard-
ing the quality of the obtained solutions?
It is not trivial to answer these questions since the
number of st-numberings of a given graph varies ac-
cordingly to its structure and characteristics. For a
complete graph consisting of n vertices, after the st
edge is chosen each vertex without a label can be a
candidate to receive the next label, thus it is trivial to
show that there are n! possible st-numberings for such
graph. It is easy to see that for that case different st-
numberings does not affect the solution since every
vertex is adjacent to every other vertex, however we
can use n! as an upper bound to the number of possi-
ble st-numberings.
Therefore, given the large st-numbering possibil-
ities space and knowing that different st-numbering
affects the quality of the solutions obtained by the
VD-PLANARISE, we decided to use a search tech-
nique to refine the solutions. For additional details
regarding the VD-PLANARISE algorithm we recom-
mend the work of (Constantino et al., 2011) and (Pin-
heiro et al., 2012).
5 THE EVOLUTIONARY
ALGORITHM
MAVD-PLANARISE
As the VD-PLANARISE algorithm possesses linear
time complexity, its use as an objective function for
optimisation techniques is efficient enough. Know-
ing that it is possible to run a st-numbering algo-
rithm with linear time complexity and that the st-
numberings possibilities space is too large to enumer-
ate, the use of an enhanced mechanism to search a
large solution space is viable. Hence we propose the
MAVD-PLANARISE.
The objective of the MAVD-PLANARISE is to
search for the best parameter setup for both the st-
numbering and VD-PLANARISE algorithms. The
MAVD-PLANARISE is defined over the basic struc-
ture of a memetic algorithm, consisting of a genetic
algorithm (Goldberg, 1989) and a local search. The
individuals are defined in a structure that contains a
copy of the adjacency structure of the graph to be pla-
narised, a (s,t) edge to be used in the st-numbering
algorithm, a st vector containing the st-numbering
of the graph over that setup and the fitness value
(number of removed vertices) calculated after the st-
numbering.
The chromosome of an individual is a copy of the
adjacency list of the given graph. Let G be a graph
and c be a chromosome; c consists of n genes where
n is the number of vertices of G. Each gene g
k
is the
adjacency list of the vertex v
k
.
Every time an individual is generated - over the
initial population generation or by crossover - the
st-numbering algorithm is applied over its adjacency
structure using the individual selected (s,t) edge with
the purpose of obtaining the st-numbering, which is
stored in the st vector. After the st-numbering is ob-
tained, the fitness value is then calculated using the
VD-PLANARISE algorithm. Only after that procedure
an individual is ready for selection and crossover.
The MAVD-PLANARISE uses a fixed size popula-
tion with a random initial generation of each individ-
ual, copying the adjacency structure of the original
graph and randomly swapping the vertices order of
each vertex list. During the selection and renewal of
the population, we opted for using an elitism system
(Goldberg, 1989) of the 10% best solutions being kept
on the population. In order to improve the selection
ICEIS2014-16thInternationalConferenceonEnterpriseInformationSystems
468
chances of less fit individuals and avoid stagnation,
the algorithm also utilises a linear scaling (Goldberg,
1989) technique to calculate the fitness value.
Regarding the selection process, after defining the
elite, the algorithm uses the roulette method (Gold-
berg, 1989) to select the pairs for crossover. The se-
lection process using the roulette method uses the fit-
ness value t
k
= f
scaling
( f
O
(k)) of each individual of
the actual population and the total value t
s
=
∑
n
k=1
t
k
,
where n is the number of individuals of the popula-
tion. After calculating those values a random value r
is picked where 1 ≤ r ≤ t
s
and the algorithm selects
the individuals that belong to the range of the sum of
the picked number.
The crossover mechanism of the proposed algo-
rithm is composed by two steps. The first one is a
regular uniform crossoveroperation (Goldberg, 1989)
and the second one is what defines the method as
a memetic algorithm, a local search to find the best
(s,t) edge to be used by the st-numbering algorithm.
After the chromosomes are generated and before cal-
culating the fitness, every new individual has a small
chance of suffering a mutation. Let this chance be α.
The mutation process uses the flip technique where
one gene is randomly raffled and all the adjacency list
of that chosen gene is shuffled.
After the crossover, the MAVD-PLANARISE run a
greedy local search procedure for each individual on
the neighbourhood of the (s,t) edge with the purpose
of choosing the best edge for that graph structure. The
procedure begins its search by picking up the best
(s,t) edges from the individual parents. The choosing
of this edge is made during the crossover and the pro-
cess consists of finding the best st-numbering given
the edge (s, t) and the inverse (t, s) of each parent
and the adjacency structure of the generated individ-
ual. Only then the greedy search starts by the selected
(s,t) edge, its st-numbering and the resulting num-
ber of vertices of the graph planarisation using that
setup. For each vertex v adjacent to s the GREEDYST-
SEARCH algorithm generates a st-numbering using
the edge (s,v) as a temporary (s, t) edge and planarise
the graph using the VD-PLANARISE algorithm. If
the resulting planar subgraph has more vertices than
the one with the original (s, t) edge, then the vertex
v replaces the vertex t in the (s,t) edge. The greedy
search also executes the same procedure for the edge
(v, s) and in case the obtained planar subgraph has
more vertices than the original one, (s,t) := (v, s) and
the GREEDYST-SEARCH algorithm return its recur-
sive call over the new (s,t) edge. The algorithm ends
when it cannot find a better solution.
Figure 3: Results for C
n
×C
m
graphs.
6 RESULTS
We tested the algorithms on two types of graphs;
cartesian graphs, for they possess symmetric and
cyclic characteristics among its vertices and edges
and randomly generated graphs. For each n, where
3 ≤ n ≤ 10, we generated ten C
n
× C
m
graphs with
m evenly spread through [n, 25], hence obtaining 80
graphs. As for the random graphs, we generated 200
graphs, with |V| evenly distributed such that 30 ≤
|V| ≤ 75 and for each pair of vertices we set an edge
with a probability of δ, where each graph was given a
random δ value such that 0.25 ≤ δ ≤ 0.75.
As for a measure, we tested the VD-PLANARISE
for every possible st-numberingusing an enumeration
algorithm. The comparative of the quality of the solu-
tions was made between the average and the best so-
lutions obtained by the VD-PLANARISE, and the so-
lution obtained by the MAVD-PLANARISE. Regard-
ing the parametrisation of our proposed memetic al-
gorithm, we defined a population of 100 individuals,
and a variable mutation rate α proportionalto the pop-
ulation’s stagnation, such that 0.05 ≤ α ≤ 0.15. For
each graph we run the algorithm three times and used
the second better result.
Figure 3 presents the chart with the results ob-
tained from the tests on the C
n
× C
m
graphs. The
first interesting observation is regarding the dis-
crepancy between the average solutions obtained by
the VD-PLANARISE and the solutions found by the
MAVD-PLANARISE, meaning that the different st-
Figure 4: Results for random graphs.
AnEvolutionaryAlgorithmforGraphPlanarisationbyVertexDeletion
469
Table 1: Summary of the experiments.
numberings in fact have great impact over the qual-
ity of the solutions obtained by the planarisation al-
gorithm. Furthermore, we can conclude that for this
special class of graphs, the algorithm performs well
enough to, in every test case, improve the quality of
the obtained solutions.
Figure 4 shows the results obtained from the tests
on random graphs. Again, we can observe that the
proposed metaheuristic was able to search the solu-
tion space and find better solutions.
Table 1 presents a summary of the experiments.
For cartesian graphs, we can highlight that the so-
lutions obtained by the MAVD-PLANARISE are very
close to the optimal solutions of the VD-PLANARISE,
overall only 0.7% worse, hence proving the quality of
the proposed algorithm for this type of graphs. For
the random graphs, it can be seen that the perfor-
mance of the MAVD-PLANARISE is superior to the
optimal solution found testing all the st-numberings.
This happens because the algorithm not only search in
the space of possible st-numberings, but also changes
the visiting order of the vertices, which affects di-
rectly the PQ-trees algorithms and therefore the VD-
PLANARISE.
The table also presents the mean of the improve-
Figure 5: Time curve for the MAVD-PLANARISE.
ments achieved by applying the metaheuristic and
comparing it to the average solution obtained by the
VD-PLANARISE. We can observe that as the size of
the graph increases, so the improvement of the solu-
tions obtained by the metaheuristic decreases. This is
expected as the search space increases and the prob-
lem gets more difficult. Nonetheless, the algorithm
performs similar both on cartesian and random graphs
as the overall improvements for cartesian graphs was
40.07% with a standard deviation of 14.682% and for
random graphs with similar size range as the carte-
sian graphs was 43.252% with a standard deviation of
8.726%.
The algorithm execution time is shown by the Fig-
ure 5 using different sized graphs in terms of number
of vertices and edges (|V| + |E|) and the execution
time in seconds. It can be noted that the algorithm
has a polynomial time performance, with a slightly
non-linear growing curve.
7 CONCLUSION
This work presented an evolutionary algorithm for
graph planarisation - MAVD-PLANARISE - which is
based on memetic algorithms. As the planarisation
heuristic algorithm, the proposed algorithm applies
the VD-PLANARISE which uses the vertex deletion
operation to obtain a planar subgraph. To the best of
our knowledge this is the only algorithm found in the
literature which optimises the number of vertices to
be removed for the process of graph planarisation.
It is important to emphasise that in the literature,
no linear complexity algorithm that planarises a
graph by removing vertices can be found as the use
of the vertex deletion operation is not frequent. Note
that the proposed algorithm finds an induced planar
subgraph from bi-connected non-planar components.
However it is possible to find a tree (or a forest in case
the graph is not connected) of bi-connected compo-
nents in linear time complexity and build an induced
planar subgraph using successive applications of the
algorithm VD-PLANARISE. Hence the algorithm
proposed here represents a sound and novel approach
to graph planarisation using the vertex deletion.
The proposed MAVD-PLANARISE approach
searches the planar solution space using different st-
numberings and obtains good results as it was shown
in section 6. Although the algorithm is capable of
refining the solution, there are no guarantees that by
just altering the st-numbering, the VD-PLANARISE
can obtain the optimal solution (as expected for a
heuristic approach). The MAVD-PLANARISE not
only searches in the st-numbering space, but it
ICEIS2014-16thInternationalConferenceonEnterpriseInformationSystems
470
also runs a local search on each individual, further
improving the results.
Future research include improvements on the
memetic algorithm in order to investigate a wider
search space, not just the one provided by the VD-
PLANARIZE. One option is to use the final solution
of the MAVD-PLANARIZE as a starting point to
another search procedure (such as simulated anneal-
ing, GRASP, VNS, PSO, etc) that does not rely on
the VD-PLANARIZE, but instead search in different
neighbourhoods. Another option is to adapt such
neighbourhoods to the local search procedure already
presented in this work. In any case, investigating
a wider range of neighbourhoods could potentially
improve the quality of the obtained solutions.
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