Algorithms and Complexity by Herbert S. Wilf

By Herbert S. Wilf

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Example text

Fix some vertex of the graph, say vertex v∗ . Let’s distinguish two kinds of independent sets of vertices of G. There are those that contain vertex v∗ and those that don’t contain vertex v∗ . If an independent set S of vertices contains vertex v∗ , then what does the rest of the set S consist of? The remaining vertices of S are an independent set in a smaller graph, namely the graph that is obtained from G by deleting vertex v∗ as well as all vertices that are connected to vertex v∗ by an edge.

There is, however, only 1 unlabeled graph that has 3 vertices and 1 edge, as shown in Fig. 8. Fig. 8: ... but only one unlabeled graph Most counting problems on graphs are much easier for labeled than for unlabeled graphs. Consider the following question: how many graphs are there that have exactly n vertices? Suppose first that we mean labeled graphs. A graph of n vertices has a maximum of n2 edges. To construct a graph we would decide which of these possible edges would be used. We can make each of these n2 decisions independently, and for every way of deciding where to put the edges we would get a different graph.

How long would it take you to calculate that number for such a graph G? How would you do it? 6. Write out algorithm maxset3, which finds the size of the largest independent set of vertices in a graph. Its trivial case will occur if G has no vertex of degree ≥ 3. Otherwise, it will choose a vertex v∗ of degree ≥ 3 and proceed as in maxset2. 7. Analyze the complexity of your algorithm maxset3 from exercise 6 above. 8. 4) to prove by induction that P (K; G) is a polynomial in K of degree |V (G)|. Then show that if G is a tree then P (K; G) = K(K − 1)|V (G)|−1.

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