Chapter - III

Adopting Topological Graph Theory to Traffic Management Problem

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Chapter-III

Adopting Topological Graph Theory to

Traffic Management Problem

Graph Theory deals with set of Vertices and Edges and relation of

incidence line connecting vertices is called an Edge. The vertices denote

starting and ending point of commuting, and the path taken by them is

represented by the Edge. A special feature of Graph theory is that, the shape

of the edge connecting two vertices is immaterial (is flexible- and can be

long or short, its shape can be varied) without altering the property of the

Graph, We are more concerned with incidence (to vertices) than the shape

of the edge. The direction of commuting is indicated by an arrow. The

Graph showing direction, is called Digraph while the one without direction

is referred to as a Graph. These features of Graph Theory are relevant to

Mathematical Modelling of Traffic Management Problem. The feature of

representing the same by means of binary relation is useful to understand

and interpret it without ambiguity. We represent starting and ending points

by Vertices and the path followed by edge (with direction). The point

where edges intersect (cross one another), which represents a conflict point,

which is responsible for accident or jam. The Crossing Number of the

graph denotes the number of conflict points encountered. A Crossing

Number ZERO implies that there are no edge crossings (conflict points)

and the Graph is called Planar, that traffic can be operated without any need

for bridges, flyover, underway or walkway, bypass etc. In simple terms,

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there is no need for engineering support/infrastructure of over-bridge, sub-

way, fly-over.

One or more edges having same pair of vertices as end vertices are

called parallel vertices. If more than one edges associated with a pair of

vertices, are referred to as parallel edges or multiple edges. A graph with

undirected edges is called Graph. When associated with direction it is

referred to as Digraph.

A Graph that has neither self-loop nor parallel edges is called a

Simple or General Graph. In Traffic Management problem no self-loop are

encountered. However, parallel edges with changed direction are very

relevant. A feature of Graph Theory, that the edges can be drawn as we

need (can be varied or altered); it can be straight or curved; long or short.

Altering the Graph would not alter its properties (planarity/incidence).

If a vertex is an end for an edge, then the vertex and edge are said to

be incident to one another. Two edges are said to be adjacent if they are

incident on a single vertex. The number of edges incident on a vertex is the

degree of the vertex.

Crossing/intersecting edges : Two edges may cross (intersect) at a point (other than vertex) in

Graphs are to be understood as edges be in different planes and thus have

no common point, if at all they are in same plane, a conflict will arise at

the point of intersection, resulting in accident for the traffic following the

edges. The Thickness of Planar Graph is ‘ZERO’ .

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The Graph with edges crossing is termed as non-planar. Thereby,

the Graph cannot be embedded on a Plane and the Graph requires more

than one place for embedding. The number of planes needed for embedding

the graph is called “Thickness of the Graph” Since, the edges represent the

path of Vehicle or Pedestrian, the point where the edges (paths) cross has

significance in Traffic Management, on it is the point where accident

occurs. The point where edges cross is referred as Conflict point in Traffic

Management. More number of conflict points indicates to more scope for

accidents. Accident is encountered when vehicles of both edges try to

occupy the point of intersection of edges.

A Digraph that represent the 4-leg intersection and the Pedestrian

Crossing, record 16 and 4 conflict points respectively. To minimize scope

of accidents, we have to eliminate the conflict points from their respective

graphs. That is to make the Graphs Planar or embeddable on one plane.

To control accidents, traffic is regulated using traffic signals which

permit 4 leg traffic by turns in four phases. Thereby, commuters of each

leg are stopped for a duration of 3 phases, before passing the intersection.

Similarly, the traffic signals at Pedestrian Crossing operate in two phases,

thereby by Pedestrian and Vehicular commuters, pass the area, each waiting

for one phase of the signal.

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Isomorphism :

Two graphs are Isomorphic if there is a 1-1 correspondence between

their Vertices; and between their edges such that the incidence relationship

is preserved.

Thereby:

a. They have same number of vertices.

b. They have same number of edges.

c. Have equal number of vertices of given degree.

d. Incidence relationship is preserved.

A Graph G’ is said to be a subgraph of G if all the vertices and all

edges of G’ are in G.

Decompose/Partition :

A graph can be decomposed into a number of sub-graphs, if their

union makes the graph and the subgraphs are mutually disjoint.

The process of Graph Theory permits, fusing or merging of vertices

and edges. If two vertices of a graph are replaced by one new vertex, such

that every edge that is incident on either of the two or on both, is incident

on the new vertex

A set of vertices and edges constituting a walk forms a subgraph of

orid its length.

A Graph is said to be connected if there is at least one path between

every pair of vertices.

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Edge-Disjoint subgraphs :

Two (or more) subgraphs G1 and G2 of Graph G are said to be edge

disjoint if G1 and G2donot have any edge in common (they may have

vertices in common).

Vertex disjoint Graphs:

Two (or more) subgraphs G1 and G2 of Graph G are said to be

Vertex disjoint if G1 and G2 have no vertex in common (however, they

may have edges in common).

Connected Graph :

A Graph is connected if we can reach any vertex from any other

vertex by travelling along the edges. i.e., It has atleast one path between the

pair of vertices.

Two disconnected graphs may have two or more connected graphs.

Each one is called its components.

A simple graph in which there exists an edge between every pair of

vertices is called COMPLETE GRAPH.

Note:

a. Complete graph with three or more vertices can have

Hamiltonian circuits.

b. In Complete Graph there are no multiple edges/parallel edges

The edge connectivity of connected graph is the minimum number

of edges whose removal (deletion) reduces the rank of graph by one.

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The vertex-connectivity of a graph is the number of vertrices, whose

removal from Graph provides a disconnected Graph.

Adoption of Planar Graph to traffic management results in free from

accident area (intersection and pedestrian crossing) , as it is free of edge

intersections (conflict points).

Isomorphism :

Two graphs are Isomorphic if there is a 1-1 correspondence between

their Vertices; and between their edges such that the incidence relationship

is preserved.

Thereby:

a. They have same number of vertices.

b. They have same number of edges.

c. Have equal number of vertices of given degree.

d. Incidence relationship is preserved.

A Graph G’ is said to be a subgraph of G if all the vertices and all

edges of G’ are in G.

Decompose/Partition :

A graph can be decomposed into a number of sub-graphs, if their

union makes the graph and the subgraphs are mutually disjoint.

The process of Graph Theory permits, fusing or merging of vertices

and edges. If two vertices of a graph are replaced by one new vertex, such

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that every edge that is incident on either of the two or on both, is incident

on the new vertex

A set of vertices and edges constituting a walk forms a subgraph of

orid its length.

A Graph is said to be connected if there is at least one path between

every pair of vertices.

Edge-Disjoint subgraphs :

Two (or more) subgraphs G1 and G2 of Graph G are said to be edge

disjoint if G1 and G2donot have any edge in common (they may have

vertices in common).

Vertex disjoint Graphs:

Two (or more) subgraphs G1 and G2 of Graph G are said to be

Vertex disjoint if G1 and G2 have no vertex in common (however, they

may have edges in common).

Connected Graph :

A Graph is connected if we can reach any vertex from any other

vertex by travelling along the edges. i.e., It has atleast one path between the

pair of vertices.

Two disconnected graphs may have two or more connected graphs.

Each one is called its components.

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A simple graph in which there exists an edge between every pair of

vertices is called COMPLETE GRAPH.

Note:

a. Complete graph with three or more vertices can have Hamiltonian

circuits.

b. In Complete Graph there are no multiple edges/parallel edges

The edge connectivity of connected graph is the minimum number

of edges whose removal (deletion) reduces the rank of graph by one.

The vertex-connectivity of a graph is the number of vertrices, whose

removal from Graph provides a disconnected Graph.

In general, Graph Theory deals with Vertices, Edges and relations

between them. For our study we prefer to deal with Edge sets and Vertices

of the edges; than Vertices and relation of incidence represented by edges

connecting them.

Decomposition:

A graph G is said to have been decomposed into two subgraphs if

G1 U G2 = G and G1 G2 = null

If it can be represented as union empty sets, which are mutually disjoint.

Note : Every edge of G occurs either in G1 or in G2. But not in both.

(However, some of the vertices, may occur in both)

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A graph can be decomposed into more than two subgraphs,

subgraphs that are (pairwise) disjoint and collectively include every edge

in G.

Planarity of Graph, roads can be laid without bridges, underway, for

traffic control and no accident / crash occurs. Is an ideal condition for a

city.

This planar property is also called embedding/imbedding. If the

Graph of the problem can be shown as Planar, then no conflict points

occurs;

Genus - The number of handles needed on a sphere in order to

embed the Graph. (Number of flyover layers).

Thickness - The number of planar graphs required to form G. This

would assess the number of tiers the road structure

needs, to adopt the graph.

Coarseness - The Maximum number of line-disjoint non-planar

subgraphs in G.

Crossing number - The number of crossings there must be when G is

drawn in the plane.

Matrix representation of graph : There are two types of matrix

representation approaches in Graph Theory :

a. Vertex-edge incidence Matrix.

b. Adjacency Matrix

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Vertex-edge incidence matrix :

Given G be a graph with n-vertices, e-edges (and no self loops). We

can represent the same by their incidence relation, by number of vertices as

row and columns. With ‘1’ for incidence of ith vertex on j’th vertex and ‘0’

otherwise.

Properties:

a. Since every edge is incident on exactly two vertices, each column of

a has two 1’s.

b. The number of 1’s in each row equals the degree of corresponding

vertex.

c. A row with all 0’s is an isolated vertex.

d. Parallel edges in graph produce identical columns in incidence

matrix.

e. If the graph is disconnected and consists of two components, the

incidence matrix can be written in a block diagram form. Implying –

no edge in g1 is incident on vertices of g2.

f. Permutation of any two rows or columns in an incidence matrix

simply rearranged or relabeled vertices and edges of the same graph

(isomorphic).

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Adjacency Matrix (connection matrix).

A Graph without parallel edges can be represented by a Binary

Matrix, represented by number of vertices for rows and columns, taking

values ‘1’ if they are connected by edge; ‘0’ otherwise.

The adjacency matrix of a graph G with n-vertices and no parallel

edges is an n x n symmetric binary matrix.

Properties of Adjacency Matrix :

a. The entries of principal diagonal are 0-s if no self-loop is present.

b. The adjacency matrix cannot represent parallel edges.

c. If the graph has no self loops ( and no parallel edges), the degree

of vertex equals the number of 1-s (in corresponding row or

column)

d. Permutation of rows or column imply reordering the vertices.

e. A graph G is disconnected and G in two components g1 and g2 if

and only if its adjacency matrix , can be partitioned.

f. Square, symmetric, binary matrix represents adjacency matrix of

a graph.

There are no methods of recording edge crossings (of the Graph).

Hence, we propose to record them in matrix form and call it as Edge-

crossings Matrix.

Since there are 12 edges in the graph, we construct a 12 x 12 matrix

with following definition

A = [a( i,j)] where a(i,j) = 1, if i’th and j’th edges cross one another

= 0 otherwise.

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Since there are no self-loops permitted in road structure, all the

diagonal terms will be ZEROs. Since each edge crossing is reflected twice

and that the matrix is symmetric, we recommend the sub-diagonal terms be

made zero. Hence, the number of 1’s occur in the matrix reflects the

CROSSING NUMBER OF THE GRAPH.

Traffic Management of 4-leg intersection.

A 4-leg, 2-way intersection is a place where four roads meet (the

roads are referred to as legs of intersection), and vehicular commuters

exchange their route. For the purpose of the study, we assume that the

commuters reach the intersection from four directions and wait. The inward

commuters will take the second lane of the leg.

The vehicular Commuters interchange the legs at the intersection.

Hence, the point where they stand before and after exchange of legs, are

denoted by vertices, the path followed the vehicle (road) is represented by

an edge. The point where the edge cross one another denote a “Conflict

Points”, which are responsible for accidents and traffic jam.

The Graph of the intersection is given below:

 

It h

The

has 8 vertic

e Matrix re

es, 12 edge

T

epresentatio

es and 16 c

The Graff o

on by mean

conflict poi

of the Inter

ns of edges

ints.

rsection

s is as follo

ows:

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I

7

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

On omitting the Sub-Diagonal terms we get:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Similarly, we draw the Graph of Pedestrian Crossing as follows:

The Edge Crossings Matrix is as follows:

Edge Crosing Matrix of the Graph

1 2 3 4

1 0 0 1 1

2 0 0 1 1

3 1 1 0 0

4 1 1 0 0

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On Omting sub-diagonal terms

1 2 3 4

1 0 0 1 1

2 0 0 1 1

3 0 0 0 0

4 0 0 0 0

And find its Crossing number to be is four and thickness one shows

the need for two planes to embed the Graph. As Road-over Bridges,

underways are not cost effective and takes large time and money for their

installation, we have tried to convert the Pedestrian Crossing Graph into 2-

connected Graph (Vertex Connected, two planar isomorphic Graphs). And

adopted to the Pedestrian Crossing, to be used in two phases. Which made

the crossing number of the Graph ZERO.

A note on Signal operation at 4-leg intersection – loss commuter time

and fuel loss due to idling at traffic signals, intensity and duration of

exposure to vehicular emission:

The Signal at 4-leg intersection operates in 4 phases. Allow one leg

commuters to pass in each phase, while other three leg commuters are

asked to wait. Thereby at any point of time, 3 leg commuters are present at

the intersection. Each commuter spends a time equivalent to 3 phases of

the traffic signal. As vehicular commuters donot switch off the engine, the

vehicles release exhaust during idling .

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The following are the observations:

1. The Loss of time at traffic signals is equal to 3 phases of the

signals.

2. The Loss of fuel (during idling) is for 3 Phases of the signal.

3. the passerby are exposed to vehicular exhaust of commuters of

three legs (intensity), duration of 3 phases of the signal.

A note on Signal operation at 4-leg intersection – loss commuter

time and fuel loss due to idling at traffic signals, intensity and duration of

exposure to vehicular emission:

The Signal at Pedestrian crossing operates in two phases. Allows

Pedestrian and Vehicles alternately. Thereby the following are the

observations:

1. The loss of time at traffic signals is equal to one phase of the

signal.

2. The loss of fuel (during idling) is for one phase of the signal.

3. The passerby are exposed to vehicular exhaust of vehicular

commuters, for a duration of one phase of the signal.