Chapter - II

Overview of Traffic Management Problems, Road Structures & Traffic Flow Models

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Over view of URBAN ROADS STRUCTURE and traffic management needs

A man is known by the company he keeps. Similarly a city or a town is

admired by its road system. A well-designed road pattern connecting the

various parts of town adds color to civil life. The urban road system

facilitates free flow of traffic with safety.

The efficiency of an urban area is greatly influenced by the urban

infrastructure of roads together with public services. It is usually found that

in many urban areas, the supply of roads has not been increased in line with

demand, the roads becoming heavily congested and parking space scarce.

The automobiles of various types have become more popular for their high

speed and flexibility. The invasion of such automobiles has reached an

alarming stage in most of the towns and cities of today.The development of

road system has lagged behind and it has resulted into a number of traffic

problems.

OBJECTS OF URBAN ROADS:

The objectives of urban roads are:

a) To facilitate commutation of men and materials between the

various centres of the town.

b) To provide air and light to the properties situated on their edges

and

c) To provide space for laying the public utility services like water

mains, drainage pipes, electric cables, telephone lines, etc.

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A Good City Road should have

a) Amenities such as shady avenues, parking places, enough lighting,

etc.

b) Safety to the vehicles and pedestrians, provision such as footpaths,

traffic signs, etc.

c) Be cheap and durable

d) Be resting on the unyielding soil.

e) Have good alignment and visibility

f) Possess easy gradients and smooth curves

g) Possess well-designed junctions

h) Remain dry condition

i) Congestion of traffic is brought down to the minimum possible

extent

j) Impervious and impermeable to the rain water

k) Its width and camber should be proper

Classification Of Urban Roads

The urban roads are of 5 types:

1. Arterial roads,

2.Sub-arterial roads,

3.Local roads,

4. Streets,

5. Pathway

1. Arterials roads: The road which connects the town to a state highway

or a national highway is termed as an arterial road. It passes within the

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city limits and carries great masses of traffic between different parts of

the town. The arterial roads are ring roads, by-pass roads, and they allow

free movement of fast traffic at high speeds. These roads are also kept

clear of all obstructions such as frontages of buildings, parking places,

loading and unloading areas. The junctions on arterial roads should be

properly designed and it is desirable to provide minimum number of

junctions on the arterial roads.

In existing cities, the arterial roads maybe missed up with the local roads.

Record of high incidence of road accidents, is seen as they are connected to

central area of the town. The purpose of arterial roads are

a) The movement and safety from through traffic

b) Are designed to carry heavy traffic and the other types of roads are

provided to carry light local traffic.

c) The through traffic is not allowed to pass through the residential

areas. Hence thereby discomfort faced by the residents is minimized.

2. Sub-arterial roads: These roads are also called major roads or

secondary roads, within city limits connecting important town centers.

These roads are meant for slow moving traffic over short distances. The

intersections on these roads are monitored with traffic signals.

3. Local roads: These roads are also called minor roads. They facilitate

traffic from various parts of the town lead to minor or major roads, is

non-vehicular traffic. The local roads are in no way connected to arterial

roads separated by a bridge so that one goes over the other.

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4. Streets: A road which serves the need of an internal portion of a local

area is called a street. It serves a residential, commercial or industrial

area depending on social, commercial and other requirements of the

local community.

5. Pathways: Are pedestrian ways and for access leading to individual

shops or premises. The vehicular traffic is not allowed to move on the

pathways.

TYPES OF STREET SYSTEMS: There are eight patterns or street

system

a Rectangular street system

b. Rectangular with superimposed diagonals street system

c .Concentric street system

d. Radial street system

e. Combination of radial and rectangular street systems

f Irregular mediaeval street systems

g. Topographical street system

h Combination of rectangular and irregular street systems

a. Rectangular street system: In this pattern, the streets are usually

of equal width and they cross at right angles (Fig. 1) It is also

known as grid-iron or checker board pattern. This pattern is

suitable for flat or level towns. This pattern adopted in

Philadelphia (U.S.A) and Jaipur (India)

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Fig.1

The advantages of Rectangular Street are

i) If a series of continuous parallel streets are provided, the traffic

capacity improves.

ii) Offers comfort and convenience due to regularity

iii) The intersections are simple in nature.

iv) It results in economic construction as the area between the streets can

divided into suitable rectangular plots.

v) The wastage of land is minimized as there are no irregular portions

left out.

Following are the disadvantages of this pattern:

There will be many intersections, hence more chances of road accidents.

a. Not suitable for uneven topography, leads to inconvenience and

discomfort.

b. Because the roads are straight the vistas are open and there is no

variety.

c. There are no short-cuts to reach to the central shopping or other

business centers.

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2. Regular with super imposed diagonals street system: This

arrangement is an improvement on grid-iron pattern. The diagonal

streets provide direct access to distant parts of the town.

The points where diagonals meet are utilized for park, garden, memorial,

etc. However, a dangerous intersection is formed at the crossing point of

diagonal-street and rectangular street. Washington, (the capital of U.S.A) is

the outstanding example of this road pattern, is one of the most beautiful

cites of the world, is an example for well- planned city. Its natural

surroundings are attractive. Salient features are as follows:

i) The circles, squares and other open spaces at junctions of streets and

avenues can be profitably utilized as sites for gorgeous statues,

monuments and fountains.

ii) The public buildings are provided with enough open spaces all

around them.

iii) The streets are very wide and are well planted with trees.

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3. Concentric street system: In this pattern roads are connected by

Radial Roads. If the radial roads are not fully developed, they will

overload the central part of the town where they intersect. Moscow

(USSR), Vienna (Austria), Paris (France) and Frankfurt-Reiderwald

(Germany) have followed this pattern.

4. Radial Street System: In this pattern, the Ring Roads spring from a Central Civic Centre like spokes of a wheel , this is to highlight importance of civic centre which may be Palace, Market, Public Office Building, Historical Monument, Beauty Spots etc., The Radial pattern is like a large blocks of Trapezoid, hence more land is required to design buildings which is uneconomical. Karlsruhe (Germany) has followed this pattern.\

5. Combination of Radial and Rectangular Street Systems: In this

arrangement, the radial streets are combined with rectangular streets

where Topographical features permit taking benefits of both the

patterns. London (U.K.), Canberra (Australia), New Delhi (India)

has adopted this pattern of road system.

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6. Irregular Mediaeval Street System: Some of the old cities of

middle ages, developed in an irregular way without any definite plan

of the street system followed this pattern. The growth of the town or

city is unmethodical and unexplainable, not suitable for modern

traffic. The towns Nurenberg (Germany), Nasik and Varanasi

(India) are of this type.

7. Topographical Street System: To take advantage of natural

topographical features of the town such as River, Lake, Sea-view,

hillock etc., is more or less like microscope view of cell structure of

vegetable or animal tissue. Here the streets are not continuous and

the width not uniform throughout. The size and shape of open

spaces are irregular, with number of curves here and there. Pleasing

and attractive due to surroundings. Letchworth and Bournemouth

(U.K.), Montreal (Canada), Amber and Vijaynagar (India) are

examples of this pattern.

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8. Combination of Rectangular and Irregular Street Systems: This

is a combination of rectangular pattern and irregular streets. The

Rectangular layout is of recent development of town, the irregular

being earlier portion of the town, since it is difficult or too costly to

improve the existing irregular street system, this combination is

availed. This pattern is followed at Edinburgh (U.K.)

By-Pass Roads and Through Road:

When a High-way Road or Through Road passes through congested

portion of the town, there will be sharp reduction in the speed of vehicles

and smooth flow is hampered. To maintain easy flow of traffic and to offer

convenience to commuters, a by-pass road is advised.

At suitable points, the High-way commuters are diverted by means

of by-pass road from one side of the town, to reach other side of the town,

without passing through busy centers. This helps in minimizing accidents at

the Busy Centers of the town.

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Outer and Inner Ring Roads:

The purpose of outer ring road is to divert the through traffic which

has no business in the town (like by-pass) for use of High Way traffic

reaching the town from all direction, which has no business in the town.

All classes of wheeled traffic will be flown on the outer ring road, and

pedestrian are barred. The Outer ring road should be barred for all

development and it should be designed as to compensate for a longer route

of unbroken passage at high speed. The purpose of inner ring road is to

facilitate traffic to avoid city centre, when they have no business in the town

centre. This helps in relieving congestion at town centre. The inner ring

road should have minimum junctions and facilitate faster passage than at

City Centre.

Expressway:

Expressway is an arterial road on which fast urban traffic can go

speedily and safely. It allows through traffic with full or partial controls,

with grade separation at intersections. The design is such that the gradient

and smooth curves.

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They are connected to the streets of the town for fast access to

destination like Airport. The cost of Expressway is prohibitive, can only be

availed when the load of traffic justifies.

Freeways:

When widening of arterial Roads do not result in relief of congestion,

the Freeway approach for use of commuters to and from neighboring

buildings are opted. This would relieve the Arterial roads of congestion.

The Freeway is only a high class arterial road having controlled access,

while the express way is of fully controlled access.

Traffic Management:

Deals with regulation of traffic, controlling and guiding the traffic,

flow of traffic at junctions, parking areas, traffic survey interpretation,

engineering aspects of road side facilities and also improvement of traffic

facilities in towns. The subject matter of Traffic Management covers the

following aspects:

a. To achieve smooth and easy flow of traffic at intersections.

b. To solve specific problems pertaining to traffic.

c. To provide safe, convenient, rapid and economic transport of Men

and Material.

d. To improve speed of commuting.

e. To improve carrying capacity of roads.

f. To provide safety of pedestrian and vehicles.

g. To minimize delays in commuting.

h. To control road accidents.

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i. To minimize traffic congestion.

j. To control cost of traffic management.

Other functions are, to assess and control nature and number of road

accidents, study route followed by Public Transport Services, Volume and

variations of Pedestrian, conditions of road surface, traffic rules and

regulations in force.

Traffic Congestion:

In Urban Centers both Pedestrian and Vehicles use the road, which

results in overcrowding on roads; steep curves and gradients or ill

maintained roads. Similarly, interaction between slow moving and fast

moving vehicles, stopping or taking turns results in congestion. There

could be several reasons for such congestion:

a. Faulty intersection, narrow roads, steep curves and gradients, poor

upkeep of vehicles.

b. Haphazard parking,

c. Plying of vehicles in conflicting directions.

d. Absence of civic sense and lack of strict enforcement of traffic rules

and regulations.

e. The streets which are originally designed for slow or fast vehicles are

used interchangeably or used by wrong group, or by all.

The disadvantages of traffic congestion are:

a. The vehicles emit pollutants which are hazardous to human and

animals.

b. Road accidents increase.

c. Driving through congested area is tedious for drivers.

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d. Loss of safety to Pedestrian

e. There will be more wear and tear on vehicles.

f. The drivers are to sit idle and results in loss of productive time.

g. Wastage of fuel due to low speeds

Several measures are taken to minimize congestion:

a. Shifting of bus-stops to suitable palaces

b. Widening of roads.

c. Dividing traffic into homogenous groups for plying.

d. Staggering the working hours of school, cinemas, and offices.

e. Commuter education by announcements, films videos, slide shows,

TV programs.

f. One-way traffic.

g. Restricting right turn.

h. Exclusive times for Pedestrian, Personal Vehicles and goods

vehicles.

i. Exclusive use by a group of commuters (fast/slow driven)

j. By-pass roads, ring roads,

k. Prohibition of U-turn.

l. Prohibiting loading/unloading of goods.

m. Restricting/controlling Pedestrian crossing.

Traffic Control:

The purpose of traffic control methods is :

a. To develop free and rapid flow of traffic on the roads and streets in

towns.

b. To prevent rapid flow of traffic this may cause undue damage to

others.

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There are two approaches to traffic control:

a. Constructive Measures: design of junctions, number of traffic lanes,

layout of curves, design of junctions.

b. Restrictive Measures: Drafting traffic rules and regulations, installing

traffic signs and signals. Penalizing defaulters.

Road Junctions, Intersection or Interchanges:

The places where two roads meet or cross one another, at same or

different elevation are terms as interchange. They are to be properly

designed for safety of vehicle and pedestrian. The intersections are to be

planned taking into consideration the following aspects:

a. Degree of traffic congestion.

b. Means for traffic control

c. Funds availability

d. Number of intersecting streets or roads.

e. Volume and nature of traffic.

f. Capability and limitations of drivers.

g. Cooperation of pedestrian.

Whatever, may the patterns of road system adopted, there are roads

and intersections, most popular being 4-leg intersection, and two way traffic

is common. The traffic jam or accidents are between Vehicle to Vehicle at

intersection or Vehicle to Pedestrian. It is the point at which their path

crosses that accidents occur. The objective of this work is to minimize or

eliminate the conflict points, so that roads can be relieved of accidents. So

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that smooth, safe, economic mobility of goods, vehicles and men will be

possible.

Characteristics of Traffic Flow:

Having dealt with Road Structure and its constituents, the

intersections and pedestrian crossings, we now deal with the characteristics

of traffic flow.

Traffic flow is a complex phenomenon constitutes of flow of

vehicles moving individually or in groups on a road, involving a human

element (driver’s behavior) controlling the vehicle. It also involves vehicle

dynamics viz., pickup and stopping time and distance, ability to take

turning, u-turn or stop. Designing and operating transportation system

efficient and safe is the need of the day.

General features of traffic flow:

Traffic constitutes vehicles transporting men and goods. Urban areas

are places of industry, employment education, medical facilities and

recreation. Thereby, commuting for the purpose of work is part of living.

Hardly any day passes without commuting. Speed at witch vehicles

commute plays a significant role in country’s economy. Time is money,

speed saves time. Wastage of time results in increase cost of living. As

Traffic flow increases, speed decreases for various reasons. Speed also

decreases when vehicles bunch together for one reasons or the other, and

driver’s discretion comes into play. Thereby, one vehicle forces others to

alter the speed, thereby congestion or jam occurs. When drivers do not take

right decision or vehicles do not cooperate with one another, accidents

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occur. At present there is no unified theory of traffic flow. The knowledge

available is much empirical and less scientific, to be applied for sure results.

More study is needed to solve the problem of congestion and accidents.

Congestion results in wastage of time and fuel; increases the cost of living.

The studies so far were based on speed, flow and density, spacing

between vehicles at different speeds, lane occupancy, clearance,

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In 1850, there were four cities I world with more than one Million

population, by the year 1950, the number has increased to 100 cities of that

size. By the year 2000, it has grown to 400 cities. By the growth of smaller

cities into bigger ones resulted in Megalopolises. Bell and Tyrwhitt in 1972

designed, Ekistic Logarithmic Scale for cities as follows:

Community Scale Population

1 t Man

2 t Room

4 t Dwelling

40 t Dwelling Group

250 t Small Neighborhood

1.5 t Neighborhood

7 t Small Town

50 t Town

300 t Large City

2 m Metropolis

14 m Conurbation

100 m Megalopolis

700 m Urban Region

5000 m Urbanized Continent

30,000 m Ecumenopolis

t – Thousands m – Million

This scale is popularly adopted for naming the Urban Centres.

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Theory of Traffic Flow:

The Theory of Traffic Flow is a mathematical study of movement of

vehicles over road net-work. It is a mathematical approach to define,

characterize and describe different aspects of vehicular traffic. It deals with

measurement of characteristics with a view to understand traffic behavior.

Gives comprehensive knowledge of vehicular traffic, offers techniques for

control, regulation and management of traffic. It has evolved from various

branches of knowledge. Statistics, Applied Mathematics, Psychology and

Operations Research. Newer and Newer theories are emerging.

Relationship between speed, flow and concentration, their relationship.

Car-following theory

Hydro-dynamic analogy.

Road traffic inevitably causes congestion. Queue and delay occur in

congestion. Distribution of cars linearly along a road, study of headways

and gaps between vehicles. Design of intersections and traffic management

at signals

The Two Model s of Traffic Flow are dealt in vogue. They are

Deterministic and Random Variation Model. The Deterministic Model

involves stochastic process, with random variations in vehicle and driver

characteristics and their interactions ignores or averages chance variation.

Hence, for a input given, produces an exactly predictable output. Hence it

is termed as Deterministic Model.

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In Random Variation Model of Traffic Flow, designed by Lay in

1986, random variation is incorporated, offers probabilities or chance

variation, thereby the output involves variance.

Categories of traffic flow

The vehicle flow is classified into two categories viz., Uninterrupted

flow and Interrupted flow. Uninterrupted flow can occur on facilities that

have no fixed elements of traffic signals, external to traffic stream that may

cause interruption to traffic flow. The traffic flow conditions are the result

of interaction between vehicles in traffic stream and between vehicles and

Geometric Characteristics of the guide way/road-way. The driver’s

intervention is limited in this type of traffic flow.

In case of Interrupted traffic flow of transport system a periodic

interruptions occur. Thereby stopping or significant changes in speed is

expected, without concerning to traffic loads and the drivers intervention is

limited.

In a congested traffic, the driver’s discretion plays a significant role,

because congestion is an internal to the traffic stream.

A well timed signaling system, an example for uninterrupted traffic

flow, to interruptions occur due to several reasons (mobility of VIPs,

Ambulance mobility) thereby the signaled system is classified as an

interrupted traffic flow system.

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Congestion, Accidents and Theory of Traffic Flow

Congestion :

Influence of one vehicle on other to speedup, slow down or stop

results in inability of the vehicle to proceed smoothly is referred as

Congestion. Often, this results in immobility of a vehicle is referred to as

Traffic Jam. When vehicles donot cooperate and driver is unable to control

the vehicle results in accident, resulting in damage of vehicle, injury and

death of the commuters. Often, the vehicle do not cooperate (speed,

breaking speed, breaking distance, heaviness of the vehicle, uncontrollable

speed) due to Laws of Motion. Traffic congestion (Jam) has become a

pressing problem all over the world especially in Urban Centres, where

proliferation of vehicles due to urbanization. Both Goods and men are to be

transported for work, education, hospital, business and recreation. Not a

single day passes without one using a vehicle.

While vehicles have its constraint, drivers too are to be well trained,

require minimum time to notice danger and take corrective action; to

control the speed or stop. Valuable time is to be saved, hence, they

overtake preceding vehicle. Drive at higher speeds, have to obey traffic

rules. They are required to maintain certain distance from his predecessor,

to avoid a crash. To facilitate overtaking, roads are widened. Thereby,

commuters overtake as they please. Speed is a factor of curvature of the

road. As curvature increases speed should decrease. Those following the

vehicle are also require to slow down, else a hit from the back results.

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While overtaking a vehicle the vehicles from opposite side are

encountered, any negligence will result in a head-on collision. These are

more fatal than other types.

The public transport buses need to stop every 500 mts. to drop or

pick-up passengers. Thereby, the vehicles following them are obstructed

and forced to overtake the bus or stop. Utmost alertness is need by those

following the bus. Often congestion occurs due to interaction of vehicles,

varied nature of drivers and vehicles do not obey the instruction of its

drivers. Due to Poor maintenance of vehicles.

To relieve congestion and accidents one approach is to widen the

road; to accommodate increased vehicles plying on the road. This is found

to have provided only temporary relief as driving at higher speeds requires

more distance to be maintained between vehicles, to avoid accident.The

extra space created is often lost and congestion is not relieved of

permanently.

Another alternative to relieve congestion is by restraint measures

viz., one-way, no right turn, no U-turn, no-parking, imposing entry fee and

heavy parking fee. These steps result in increased traffic load to

neighboring areas. Installing Traffic Signals is popularly followed as a

Accident Preventive Measure.

Then simulating future traffic density and computing average

velocity of constituent vehicles, to determine which control measure is

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adopted. Some countries have tried to reckon total time spent by all

vehicles in the traffic network, traffic density and area. The congestion

resulting traffic density, so that controls can be exercised to prevent

congestion asking commuters to take alternative route. As the time spent

increases, congestion also increases was noted. This results in Environment

costs (air pollution, fuel consumption) increase. The social costs too

(delayed delivery of goods) results.

Often Model based approaches are followed to predict future traffic

situation for adopting control strategies to ensure traffic management at

lowest cost. Since safety of commuters is of atmost importance, traffic

operation near congestion with shockwaves are being simulated to find

solution to traffic congestion problem.

Note on Theory of Traffic Flow:

There are four theories to Traffic Flow. They are:

A. Physical Theory of Traffic Flow

B. Level of Detail Theory (Micro/Macro) of Traffic Flow.

C. Deterministic and Stochastic Theory of Traffic Flow.

D. Discrete-Continuous Theory of Traffic Flow.

A. Physical Theory of Traffic Flow :

System Theory proposed by Ljung (1987) there are three types of

Traffic Modelling viz., Deductive, Inductive, Combined Models.

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a. Deductive (White Box) Model :

Physical equations represent relationship between state of traffic

system (based on Newtons Laws of Motion). This is also called measurable

states or mechanical system.

b. Inductive (Black Box) Model :

In Inductive Model, Input and output are recorded and a geneic

parameterized model is fitted to the data (traffic flow, traffic density,

measured speed over time). Here no physical relationship exists between

situation and model structure. Ho and Ioannon (1996) proposed this system

have assumed that neural network mimic the behavior of the traffic system.

c. Combined (Grey Box) Model :

This Model is a combination of both Deductive and Inductive

approaches, wherein the parameterized equation between states of the

motorway system are noted. During Inductive Phase, the parameters in the

model are tuned by fitting the input-output relation to the input-output

measurement of the traffic system.

Lighthill, Whtham (1955)

Richards (1956)

Payne (1971) have adopted these approaches.

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B. Level of Detail Theory of Traffic Flow:

Hoogendoorn and Bovy (2000) proposed two types of traffic flow

Models basing on level of detail theory.:

a. Microscopic Traffic Model.

b. Macroscopic Traffic Model.

a. Microscopic Traffic Model: This model assumes vehicles as

particles in a stream. The behavior of vehicle, by interaction between

vehicles, between vehicles and motor way are simulated.

Several conditions like speed limit, number of lanes, overtaking

prohibited are imposed. Every vehicle in the network is characterized by

some parameter describing its origin, its destination, desired velocity,

acceleration and deceleration. The vehicle parameters (Mass, acceleration;

breaking speed, breaking distance) are significant. The Driver parameters

(aggressiveness, reaction time) play full role in this Model.

The vehicles each attempt to reach their desired destination by

travelling through the network. The vehicles interact with each othr (by

overtaking slower vehicles) and with motor way netwrom (obeying speed

limits et.). Internal and external constrainsts are encountered in this Model.

In Microscopic Model there are two approaches :

i. Car following approach

In this Model, driver preserves the way his preceding vehicle drives.

Driver reacts to acceleration and deceleration of his preceding vehicle . The

accelerating ability/ the mass of the vehicle (vehicle specific parameters are

used).

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The external factors of Maximum Speed Limit, Number of lanes on

motorway specific parameters, influence the way a car follows its

predecessor.

ii. Overtaking approach:

In this approach, driver decides whether or not to overtake its

predecessor.

Vehicle properties – Mass of vehicle

Accelerating ability

Breaking ability

Driver property - Ability to sense risk

Promptness in taking corrective action.

Selection of speed while overtaking.

Motorway property. Road width,

Number of lanes.

According to Nagel (1996), Microscopic Models can be

implemented are viewed as cellular automation models, where the motor

way is viewed as a network of connected cells. The behavior of the vehicle

in the network .The network is described as ‘hopping’ from one cell to the

next. A set of rules for hopping are defined and rules influence the driver

how to operate. (No vehicle can hop a cell to the next, if the cell is

occupied).

b. Macroscopic Traffic Model:

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The Macroscopic Models use aggregate of variables (summarized

information about multiple vehicles – average speed of all vehicles present

in section of the road). It defines a relation between traffic density, average

velocity and traffic flow.

Evolution of Traffic Models:

Lighthill and Whitham (1955) and independently by Richards (1956)

– is first order Traffic Model. (Only state variable is based on Traffic

Density).

Extending this Model to cope with shock waves and stop-and-go

traffic in congested traffic situation by Newell (1993). Payne (1971) has

proposed 2nd order Traffic Model involving two state variables – traffic

density and average velocity.

Helbing (1996) has proposed a third order macroscopic traffic model

with state variables – traffic density, average velocity and variance on the

velocity.

Macroscopic Traffic Models work on aggregate variables and do not

describe the traffic situation on the level of independent vehicles, they are

less computationally intensive than Microscopic Models.

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Simulation, faster for on-line predictive control. The cost function is

aggregate variable. As the Macroscopic Model has fewer parameter to

estimate than in case of Microscopic Model it is easier to identify and tune

the Macroscopic Model. Hence Macroscopic Models are preferred Model

for optimal predictive and adoptive control.

The Traffic flow does not fit exactly to any one of the theories

discussed above hence the results are nt readily derivable. The study of 137

Countries by the Working Group on Speed Mobility solutions found that the

results are not matching investment. To control accidents, minimize loss of

time at traffic signals; minimize wastage of fuel at signals and exposure to

vehicular emission, we wish to examine the problem afresh by Topological

Graph Theory Approach.