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Intersection Graph in Traffic Control Problems International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 1, Mar 2013, 265-270 © TJPRC Pvt. Ltd. INTERSECTION GRAPH IN TRAFFIC CONTROL PROBLEMS ARUN KUMAR BARUAH & NIKY BARUAH Department of Mathematics, Dibrugarh University, Dibrugarh, Assam, India ABSTRACT A traffic control problem is studied with the help of intersection graph. The compatible streams of traffic are allowed to get overlapping arcs; which is also an intersection graph. The solution of the traffic control problem at an intersection is obtained by dividing the cliques of the intersection graph and phasing of traffic lights is done by representing the cliques of the intersection graph as signal groups. KEYWORDS: Circular Arc Graph, Intersection Graph, Interval Graph, Signal Group, Traffic Control INTRODUCTION The increase in urbanization and traffic congestion creates an urgent need to operate our transportation system with maximum efficiency [3], [8]. As traffic volumes continues to increase, the streets become more and more congested. One of the most cost effective measures for dealing with this problem is traffic signal control. Traffic signal retiming and co-ordination of existing signal have been proven to bring about substantial reductions in traffic delay, considerable energy saving and consequently huge reduction in travel time and increased safety for the public. The control measures introduced to improve traffic performance in road traffic include speed limit control, ramp metering, user information aiming at minimizing the number and the severity of accidents. On the other hand, introduction of electronics and computer systems in vehicle technologies have significantly contributed to safety and comfort. Historically, road authorities responded to the increasing demand by adding capacity, building new roads or expanding the existing ones [10]. With the high cost and constraints on building conventional infrastructure, maximising the effectiveness of the existing system including capitalising on new technologies such as Intelligent Transportation System (ITS) has become a new focus for traffic control at an intersection [9]. Given demographic trends and the growing demands for improved system performance, road authorities are changing the way they plan and operate their transport systems and are focusing more intensely on road network operations. More importantly, the transportation world has become increasingly user-driven. Traffic users are concerned with mobility and accessibility. They want transportation choices and real time information in order to make informed decisions. Improved travel time and congestion relief are desired. Greater reliability of the transportation system and the minimisation of unpredictable delays, greater safety and security are also highly expected by the traffic users or the participants. Control of traffic signals for efficient movement of traffic on urban streets constitutes a challenging part of an urban traffic control system. Traffic signal control varies in complexity, from simple systems that use historic data to determine fixed timing plans, to adaptive signal control, which optimizes timing plans for a network of signals according to traffic conditions in real time. Some of the common traffic signal operations for controlling the traffic flow are cycle time adjustment, split adjustment, where split is defined as the fraction of cycle time that is allocated to each phase for a set of traffic movements. 266 Arun Kumar Baruah & Niky Baruah Traffic control at an intersection or network is thus much more than keeping the system running. It is about optimising the system performance measures that is to adopt performance measures that will evaluate the efficiency and effectiveness of the road network thus improving the transportation system performance [6]. INTERSECTION GRAPH Let F be a family of non empty sets. The intersection graph of F is obtained by representing each set in F by a vertex and connecting two vertices by an edge if and only if their corresponding set intersects [4]. Formally an intersection graph is defined as: An undirected graph G= (V, E) formed from a family of sets, ; i = 0,1,2,... by creating one vertex for each set and connecting two vertices and by an edge whenever the corresponding two sets have a non empty intersection, that is E(G) = } For example, a family of sets = { a, b, c, d } = { c, e, h } = { e, f, g } = { h} and its corresponding intersection graph (Figure 1), Figure 1: An Intersection Graph where , , and are the sets defined above. Any undirected graph can be represented as an intersection graph: for each vertex of G form a set consisting of the edges incident to ; then two such sets have a non empty intersection if and only if the corresponding vertices share an edge. Hence the following theorem can be stated as: Theorem Every graph is isomorphic to the intersection graph of some family of sets [4]. Intersection Graph in Traffic Control Problems 267 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. The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an interval graph. If these intervals are required to have unit length, then we have a unit interval graph; a proper interval graph is constructed from a family of intervals on a line such that no interval properly contains another. 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 [5] In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph may be represented as an intersection graph, but some important special classes of graphs may be defined by the types of sets that are used to form an intersection representation of them. FORMULATION OF THE PROBLEM Let us consider a traffic intersection with traffic streams a, b, c, d, e, f, g as shown in the Figure 2. Certain streams are judged 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 [7]. The vertices of the graph are the traffic streams and two streams are joined by an edge if and only if they are compatible i.e. can be allowed to move simultaneously without any conflict. Here in this paper we shall use intersection graph for phasing of traffic lights at an intersection and show that it can be efficiently used to model a traffic control problem. As cliques of a compatibility graph represent a solution to the control problem, here we shall consider the cliques of the intersection graph as a solution and phasing of traffic lights is done by dividing the cliques as signal groups [1]. Figure 2: A Traffic Intersection with Seven Streams The above intersection can be represented by a compatibility graph G, where the traffic streams which can simultaneously move together will be joined by an edge and the streams represents the vertices of the compatibility graph as shown in the Figure 3. 268 Arun Kumar Baruah & Niky Baruah Figure 3: Compatibility Graph G of the above Intersection The above compatibility graph G can be represented as assignment of an arc of a circle to each traffic stream so that only compatible traffic streams get overlapping arcs. In traffic light phasing, we wish to assign a period of time to each stream during which it receives green light and do it such a way that only compatible streams can get green light at the same time. Figure 4: A Feasible Green Light Assignment The intersection graph corresponding to any feasible green light assignment (Figure 4), is a subgraph of the compatibility graph. It is not necessarily a generated subgraph, but it corresponds to a subgraph with same vertex set as the compatibility graph, but one with perhaps some of the edges deleted, so called a spanning subgraph. The above diagram of feasible green light assignment can be represented by intersection graph H (Figure 5) which corresponds to the arcs of the circle. Figure 5: Intersection Graph H Corresponding to the Feasible Green Light Assignment Intersection Graph in Traffic Control Problems 269 SOLUTION AND PHASING OF TRAFFIC LIGHTS The cliques of the above intersection graph are: = { a, b, e} = { b, d, e} = { c, g} = { f, g} The cliques of the intersection graph H corresponds to the phase i.e. there are four phases and if we consider the cycle time to be 60 seconds each can be allowed to move once in the cycle which is the trivial solution [2]. But a better solution can be obtained if we wish to minimise the total amount of waiting time. Let us consider the cliques of the intersection graph H as the vertices of the graph and an edge joining the vertices if there exist a non empty intersection. Thus the cliques set correspond to the family of sets. i.e. F = { , , , } where ; i = 1,2,3,4 and we associate a graph with F called the intersection graph (Figure 6). Figure 6: Intersection Graph of the Cliques Now if we want to phase these set of cliques we can put and in a set and and in another set i.e. = { , } = { , } ; where , , and are as defined earlier. Again considering the cycle time to be 60 seconds; each of the signal groups can be allowed to move for 30 seconds each which is a better solution as compared to the earlier one as most of the traffic streams are moving which reduces the waiting time.
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