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By Herbert S. Wilf

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In this chapter we’re going to work out a number of examples of recursive algorithms, of varying sophistication. We will see how the recursive structure helps us to analyze the running time, or complexity, of the algorithms. We will also find that there is a bit of art involved in choosing the list of variables that a recursive procedure operates on. Sometimes the first list we think of doesn’t work because the recursive call seems to need more detailed information than we have provided for it.

3 we show the collection of all of the graphs that the compiler might generate while executing a single call to maxset1 on the input graph of this example. In each case, the graph that is below and to the left of a given one is the one obtained by deleting a single vertex, and the one below and to the right of each graph is obtained by deleting a single vertex and its entire neighborhood. Now we are going to study the complexity of maxset1. The results will be sufficiently depressing that we will then think about how to speed up the algorithm, and we will succeed in doing that to some extent.

If we don’t have enough colors, and G has lots of edges, this will not be possible. For example, suppose G is the graph of Fig. 4, and suppose we have just 3 colors available. Then there is no way to color the vertices without ever finding that both endpoints of some edge have the same color. On the other hand, if we have four colors available then we can do the job. Fig. 4 There are many interesting computational and theoretical problems in the area of coloring of graphs. Just for its general interest, we are going to mention the four-color theorem, and then we will turn to a study of some of the computational aspects of graph coloring.

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