Traffic Flow and Circular-arc Graph

By

Abdulhakeem Mohammed

Outline

• Problem Statement• Problem Formulation• Construction Graph• Circular-arc Graph• Maximal Clique

The problem of traffic flow congestion

• Collision at an intersection of traffic lanes.• To avoid collisions, we wish to

install a traffic light system to control the flow of vehicles. • Assigns to each lane a period

of time, which the lane has the right of the way.

Formulation of the problem• Let’s consider a traffic intersection with traffic streams t, v,

w, x, y, z as shown in the Figure 2. • Certain streams are termed to be compatible with each

other in the sense that they can move through the intersection at the same time without dangerous consequences.

• The decision about the compatibility is made before time, by a traffic engineer and may be based on estimated volume of traffic in a steam as well as the traffic pattern.

• The compatibility information can be summarized in a graph called the compatibility graph.

• The vertices of the graph are the traffic streams and two streams(Vertices) are joined by an edge if and only if they are compatible i.e. can be allowed to move simultaneously without any conflict.

Figure (1) Traffic Flow Intersection

Constructing Graph

Figure (1) Traffic Flow Intersection

Figure (2) Compatibility Graph G

Intersection Graph

• The problem of characterizing the intersection graphs of families of sets having some specific topological or other pattern is often very interesting and frequently have applications to the real world.

• Well-known Intersection graphs are Interval graph, choral graph, permutation graph, circular-arc graph and so on.

• The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an interval graph.

• The intersection graph of a family of arcs on a circularly ordered set is called circular-arc graph.

• If the two ends of the line are joined thus forming a circle, the intervals will becomes arcs on the circle. Allowing arcs to slip over and include the points of connection, we obtain a class of intersection graphs called the circular arc graph which properly contains the interval graph.

Circular-arc graph • A Circular-arc graph (see Figure Below ) is an intersection graph of arcs on the circle. • That is, every vertex is represented by an arc, such that two vertices are adjacent if and only if the

corresponding arcs intersect.• The arcs constitute a circular-arc model of the graph.• Circular-arc graphs generalize interval graphs which are the intersection graphs of intervals on the

line. • Every interval graph is circular-arc graph, however, the converse is false.• Cutting the circle at some point p and straighten it out on the line, the arcs becoming intervals.• Circular-arc graphs can be used to model objects of a circular or a repetitive nature.

Circular-Arc Graph

Figure (2) Compatibility Graph G

vt

x

yw

z

Figure (3) Circular-Arc Representation for Graph G

• Each lane(vertex) assigned an arc on a circle representing the time interval during which it has a green light. Incompatible lanes must be assigned disjoint arcs.

• The circle may be regarded as a clock representing an entire cycle which will be continually repeated.

Maximal Clique

• A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. • In Another Words, A clique of graph G is a maximal clique of G if it is

not properly contained in another clique of G.

Figure (2) Compatibility Graph G

The maximal clique of G are:

k1={t,v,y} k2={x,v,y} k3={v,z} K4={z,w}

References

1. Golumbic, M. C. (1980). Algorithmic graph theory and perfect graphs. New York: Academic Press.

2. Stoffers, K. E. (1968). Scheduling of traffic lights—a new approach. Transportation Research, 2(3), 199-234.

3. Bharali, A., & Baruah, A. K. (2013). Optimal Feasible Green Light Assignment to a Traffic Intersection using Intersection Graph. International Journal of Computer Applications, 65(12).

Thank you

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