Traffic Lights, Graph Theory Problems.... 
Now, for starters, I feel that this picture must be represented Graphically and the solution does show a Graph, which is given below 
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But I am unable to understand this explanation. Could someone please explain this by perhaps giving an example? 
Edit: 
What is the equivalent graph of this figure?
 A: Not sure what part it exactly is you didn't understand:


*

*how the graph is built:


Each vertex represents a traffic lane. Since you will get an accident if cars from lanes $L_1$ and $L_2$ have green lights at the same time, you draw an edge between $L_1$ and $L_2$ on the graph. Proceed in the same way for each couple of vertices, and all constraints are summed up in the given graph.


*

*how the solution works:


From the graph, you need do find a coloration: assign a group to each vertex so that vertices in the same group can have green lights at the same time, i.e. there is no edge between two vertices in the same group.
You need at least three groups since $L_2$, $L_4$ and $L_6$ are all connected. The coloration shown solves the problem and has only three groups, so it is minimal.
A: Answering the edit:
In the graph associated to your second example of intersection:
It has 8 vertices (L1 to L8)
Its edges are:
(listing only the links that have not already been mentioned)


*

*L1 is connected to L3, L4, L6, L7, L8

*L2 is connected to L3, L4, L5, L7, L8

*L3 is connected to L5, L6, L8  

*L4 is connected to L5, L6, L7  

*L5 is connected to L7, L8  

*L6 is connected to L7, L8  
