The  traffic  control  has  to  be  managed  systematically  to  avoid  the  traffic  congestion especially in the busy city.  The efficient and systematic traffic control is studied in this research using the connectivity and compatibility graphs of traffic intersections.  From the graphs drawn, the most efficient route can be determined and the capacity of traffic flow can be maximized by finding the minimum number of edges or the minimum number of vertices.  In this paper, the 4-ways  intersection  stream  at  Jalan  Membunga  Machang  is  chosen  to  find  suitable  locations  to place sensors that are used to collect traffic data.  From the graphs obtained, there are 12 vertices identified where 36 edges were connected to it.  By using the algorithm of graph theory, four sets of  minimal  edge  control  were  determined  and  these  edges  were  validated  using  the  MAPLE softwar

Abd Ghani, Nurul Nadiah

Ismail, Mardhiyah

Mohamed, Firdawati

Salleh Huddin, Nur Shamimi

Wan Hassan, Wan Nur Hafawati

English

Universiti Teknologi MARA Institutional Repository

 
JOURNAL OF MATHEMATICS AND COMPUTING SCIENCE, 2018,  
Volume 4, Number 2, 27-33 
 
eISSN 0128-0767 © Unit Penerbitan UiTM Kelantan   27 
 
Efficient Sensor Position Selection Using Graph Connectivity 
 
Firdawati Mohamed 1*, Nurul Nadiah Abd Ghani 2 , Mardhiyah Ismail 3,  Nur Shamimi Saleh Huddin 4 and Wan 
Nur Hafawati Wan Hassan 5 
1,2, 3, 4,5 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Kelantan, Bukit Ilmu, Machang, 
Kelantan, Malaysia 
1*firdawati02@kelantan.uitm.edu.my,  
mardhiyah2264@kelantan.uitm.edu.my, nad.infinate@gmail.com, mimiey25@gmail.com, efawanhassan@gmail.com 
Abstract: The traffic control has to be managed systematically to avoid the traffic congestion 
especially in the busy city.  The efficient and systematic traffic control is studied in this research 
using the connectivity and compatibility graphs of traffic intersections.  From the graphs drawn, 
the most efficient route can be determined and the capacity of traffic flow can be maximized by 
finding the minimum number of edges or the minimum number of vertices.  In this paper, the 4-
ways intersection stream at Jalan Membunga Machang is chosen to find suitable locations to 
place sensors that are used to collect traffic data.  From the graphs obtained, there are 12 vertices 
identified where 36 edges were connected to it.  By using the algorithm of graph theory, four sets 
of minimal edge control were determined and these edges were validated using the MAPLE 
software. 
Keywords: Graph Theory, Sensor, Traffic Light. 
1 Introduction 
 
The purpose to have traffic signals is to regulate the conflicts of traffic flows at intersections.  If 
the traffic signal is managed systematically, it can gives benefit to the road users and if it 
configured incorrectly, the problem associated with it such as traffic congestion will happen.  In 
the traffic light system, the three colors red, green and yellow are chosen to represent different 
actions. Red is chosen as the color for stop because red has for centuries been used to indicate 
danger while green is chosen as the color to go because it has the color of nature and can  
represents safety [1]. 
The traffic control system was developed significantly since the first non-electric traffic light 
control was installed in London in 1868 [2].  In 1914, the electric traffic light was developed in 
Cleveland, Ohio with the red and green as the chosen color schemes for the traffic light. In 1960, 
the very first traffic controller system was then developed and was introduced in Toronto. 
The electromechanical devices were used to perform the traffic control at first [3] and was later 
upgraded to the Intelligent Transportation System (ITS).  ITS was extensively used in urban areas 
to control traffic at an intersection.  ITS is an information and communication technology that 
aims to provide the infrastructure and vehicles management to improve safety, mobility and 
efficiency from the basic management system to monitor applications and to more advanced 
applications that integrate live data and feedback from other sources. Wireless communication, 
computational technologies and sensing technologies are some of the components implemented in 
ITS.  The accurate traffic data from the traffic flow can be collected by using the semi conductor-
based controller known as sensor.  These sensors were placed at the intersections to collect the 
necessary traffic information for traffic control system [2]. 
Graph theory is the branch of mathematics concerned with networks of points connected by lines 
(vertices that are connected by the edges).  This theory is capable to solve the system of traffic 
lights as applied on konigsberg bridges by Leonard Euler in 1736.  The traffic system developed 
for this bridge was divided into four parts and connected by seven bridges.  The assumption made 
by Euler for the developed system was  everyone will only cross the bridge once [4]. 
Firdawati Mohamed 
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Traffic management is important to regulate the flow of traffic.  Congestions can occur due to the 
increase of urbanization, population growth and the number of vehicles on the road.  Based on the 
report released by The Malaysian Automative Association (MAA), 666,465 vehicles were 
registered for the year 2014, increase by 1.6% from the previous year.  Malaysia also recorded as 
the third highest car ownership globally with 93% of the households owned personal cars  and 
54% owned multiple car ownership [5]. 
The increase in the number of vehicles on the road will definitely create the traffic congestion 
especially in the cities area. This phenomena will cause the increase in the traffic accidents, 
environmental and air pollutions, the travel time and the fuel consumptions.  
2 Literature Review 
 
The traffic lights was used to regulate the flow of traffic to avoid collisions.  The arrangement of 
the traffic must be managed properly to avoid traffic congestions especially in the cities area where 
the number of vehicles on the road was doubled due to the increase in population growth. In the 
research done by [6], the signed graph was used to model the transportation problem where the 
nodes represents the destination and the edges represents the relationship among them.  Based on 
their analysis, positive (+) sign was obtained and shown the balance graph.  The result indicates 
that the traffic flow at the intersection was smooth and well planned. 
As investigated by [7], one of the problem affect the development of most city areas are traffic 
congestion.  This situation makes people late to work, the deliveries cannot be on time and cost a 
lot of money for fuel expenses.  Constructing new roads can be one of the alternative to solve the 
issue but the cost incurred will be higher and the problem cannot be solved in a short time period.  
The only solution to this problem is to use the current road network efficiently.  They introduced 
the compatibility graph corresponding to the problem and circular arc graphs to reduce the waiting 
time at junctions. 
In the research done by [8], the interval graph was used to overcome the problem at an intersection 
among the traffic users.  Since the traffic control at an intersection is crucial due to its complexity, 
the relations among the streams was firstly identified.  The streams were considered as vertices 
and will be connected by an edge if the two streams are able to move simultaneously without any 
conflicts.  The interval graph was obtained and be used to find the optimal number of feasible 
green light assignments.   
Graph Coloring Theory was used by [9] on the intersection of five fork roads.  The compatible 
graph found in this research is not the interval graph.  Therefore, the several edges were opened by 
the researchers to create an interval graph from the subgroup of the compatible graph.  Linear 
programming was also used in their research to optimize the control time for each road in the 
intersection. 
Traffic control problem at an intersection can also be modelled as a graph.  From the research done 
by [3], the cut-set of graph was used to identify the efficient route from the minimum number of 
edges.  The cut-set graph is also known as an edge control set of graph.  This graph is obtained by 
joining the traffic streams that is compatible at the intersection with an edge. The streams which 
cannot move simultaneously will not be connected to an edge. Minimal edge control represents the 
flow of traffic at the intersection and can be found by applying the algorithm. Therefore, by 
finding the minimal edge control set, the sensors can be placed in the right place and the waiting 
time of traffic users can also be minimized  
The compatibility graph was used by [10] by connecting two streams with an edge.  These two 
streams are compatible if they can move simultaneously without any accidents when the green 
lights are on. The duration for the traffic light to turn to a green light corresponds to an arc of a 
circle was considered in their research. The spanning subgraph (intersection graph from the 
 Efficient Sensor Position Selection Using Graph Connectivity 
 
eISSN 0128-0767 © Unit Penerbitan UiTM Kelantan   29 
compatibility graph) with the most maximal clique used to identify the optimum waiting time for 
all streams in a cycle [7]. 
The compatibility graph was also used by [2] in their research. They divide the cliques of the 
compatible graph as signal groups by taking into consideration the compatible streams that have 
the green light simultaneously.  The spanning subgraph which is the intersection graph 
corresponding to any feasible green light assignment was formed to calculate the waiting time at 
the intersection. 
3 Methodology 
 
A Identify the Traffic Streams 
 
In this paper, the stream at Jalan Membunga, Machang was considered. The traffic streams were 
redrawn based on the information obtained from Jabatan Kerja Raya (JKR) Machang as shown in 
Figure 1. 
 
 
Figure 1: 12 Streams at the 4-way Intersections of Jalan Membunga, Machang, Kelantan. 
 
B Develop the Compatibility Graph  
 
Compatibility graph is a graph G whose vertex set, V consist of the traffic streams and E is the two 
vertices of G are connected if, and only if, the corresponding streams are compatible [10]. Thus, 
Compatibility graph of G = (V,E). A matrix of incidence A was form by 
 




intersect  and  movements if,0
intersectnot  do  and  movements if
jk
jk
aij
,1
 
 
where k and j both are the traffic streams [11]. Hence, the matrix formed is used to draw the 
compatibility graph by using the software MAPLE. 
  
C Determine the Minimal Edge Control Set  
 
A graph G is a pair of sets  EV , , where V is the set of edges of G. An edge control set F is said 
to be minimal if any proper subset of F is not an edge control set of the graph G, Niky and Arun 
[6]. As the edge control set of a graph is not unique, therefore it is important to find the set with 
the minimum number of edges. Niky and Arun [5] found that the algorithm can be used in order to 
find the minimal edge control set by using the defnition as follows: 
Firdawati Mohamed 
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Definition 1: Graph G = (V,E) is considered to be a graph, let H be a subgraph of G  
and 𝑒 ∈ 𝐸(𝐻). We define:CH(e) = {𝑒} ∪ {𝑑 ∈ 𝐸(𝐻): d is a cut edge of H – {e}} 
Then, CH(e) is called the control of e in H. 
 
Algorithm 1: A graph G is considered and a subset F ⊆E(G) is constructed by the 
following steps: 
 
Step 1: Let F : = 𝜙 and H: = G 
Step 2: While E(H) ≠  𝜙, select any edge 𝑒 ∈ 𝐸(𝐻) 
F : = F ∪ {𝑒}  
H: = H - CH(e) 
Then, F is the minimal edge control set of the graph G. 
 
D Draw the Connectivity Graph  
 
According to [2] the connectivity graph is a graph where the removal of edges disconnect the 
compatibility graph. There are two ways to obtain connectivity graph either by removing vertex or 
removing edges. In this case study, removal of edges based on the compatibility edges is chosen. 
So, the connectivity graph can be obtained by declaring vertices and edges of the graph. The edges 
are obtained by subtracting the edges of compatibility graph with edges of minimal control set 
where the vertices are same. 
4 Results and Discussion 
 
The results obtained in this paper are shown in this section.  From the compatibility graph drawn, 
12 vertices and 36 edges are found.  The minimal edge control set is identified to find the effective 
location to place the sensor from the connectivity graph drawn.   
 
A Compatibility Graph 
 
The compatibility graph shown in Figure 2 is drawn based on matrix A  in Figure 3.  This graph 
has 12 labeled vertices and 36 edges.  The vertices are connected to each other if they are 
compatible. This compatibility graph is used to find the minimal edge control to draw the 
connectivity graph. 
 
 
Figure 2: Matrix A 
 
Figure 3: Compatibility Graph 
  
 Efficient Sensor Position Selection Using Graph Connectivity 
 
eISSN 0128-0767 © Unit Penerbitan UiTM Kelantan   31 
B Minimal Edge Control Set 
 
The set F = {(a, c), (b, c), (c, g), (c, j)} is the set of minimal edge control of the compatibility 
graph. From the set F we have (a,c), (b,c), (c,g), (c,j) as the four minimal edges control  of the 
compatibility graph. These edges will determine the location to place the sensors.  Based on the 
result obtained,  the sensors should be placed on the first, third and fourth phase at the traffic light 
streams.   Noticed that the vertex c in the set F is the deleted edge in the compatibility graph and as 
a result, the connectivity graph can be formed. 
From set F, the edge connectivity of (a,c), (b,c), (c,g), and (c,j) were proven by using  the 
“IsCutSet(S)” command in Maple to validate whether the sets obtained is true or false.  To prove 
this,  the edges that have the smallest degree of vertex is identified which are (a,b), (b,c), (c,j), and 
(c,g).  These edges are then tested in “IsCutSet(S)” of Maple Software to prove for its 
connectivity.  The result turned to be true for all the four edges. 
“IsConnected” command  of Maple Software is used to determine whether the graph is connected. 
From the Graph G, the connectivity graph obtained by deleting the disconnected edges using the 
“DeleteEdge” command.  The “IsConnected” command was then used to test the connectivity of 
the graph after removing the disconnected edges.  If the result displayed as true, the graph obtained 
is proven connected. 
 
(i) 
 
(ii) 
 
 
 
(iii) 
 
(iv) 
Figure 4: Steps in Deleting the Disconnected Edges using “DeleteEdge” Command 
Firdawati Mohamed 
eISSN 0128-0767 © Unit Penerbitan UiTM Kelantan   32 
C Connectivity Graph 
 
The graph is not connected because of vertex c. And the minimal control edges (a,c), (b,c), (c,g), 
and (c,j) obtained from the previous section  are the deleted edges. Therefore, the four minimal 
control edges were removed to get the connectivity graph as shown in Figure 5. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5: Graph Connectivity 
 
 
To get the connectivity graph, there are five possible number of traffic streams need to be chosen 
from 8, 9, 10, 11, and 12 streams. The edge connectivity of 8 streams is zero, one edge is 
connected for 9 streams, two edges connected for 10 traffic streams and three connected edges 
formed from 11 traffic streams.  In this paper, 12 traffic streams is considered for the 4-way 
intersection at Jalan Membunga that was controlled by four phases of traffic light. 
5 Conclusion 
 
In this paper, the cut-set graph was used to study traffic control problems at an intersection.  The 
four sets of minimal edge control (a,c), (b,c), (c,g), (c,j) were identified.  The locations to place the 
sensors were also determined namely phase 1, phase 3 and phase 4 and these places are the most 
effective places to collect the traffic data. 
 
The research of traffic control in  Jalan Membunga, Machang is relevant since the pedestrian road 
was also considered in this research. 
 
It is recommended to consider the minimum waiting time at the intersection using interval  
intersection graph for the future research.  
 
Acknowledgements 
 
The authors wish to express unending gratitude to Faculty of Computer and Mathematical 
Sciences (FSKM), UniversitiTeknologi MARA (UiTM) Cawangan Kelantan, Malaysia for 
their continuing support. 
 
 
 
 
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