This problem is very well known. It is called minimum vertex problem. In this problem we have a graph and the solution is a sub-set of nodes such that for any edge of the graph, either one extreme or the other, belongs to the set. And of course we want to find the minimum number of such nodes. You can think to this problem like the policemen problem, in which one policemen wants to put some police cars in several squares of the city in order to control all the streets of the city. In this case the policemen would like to know what is the minimum number of cars he need in order to cover all the streets of the city.

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