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How many cliques does it have? 15. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, 3} a triangle? 16. Let H be a labeled graph of L vertices. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, . . , L} equal to H? 17. Devise an algorithm that will decide if a given graph, of n vertices and m edges, does or does not contain a triangle, in time O(max(n2 , mn)). 18. Prove that the number of labeled graphs of n vertices all of whose vertices have even degree is equal to the number of all labeled graphs of n − 1 vertices.

Then G would contain only vertices with 0 or 1 neighbors. Such a graph G would be a collection of E disjoint edges together with a number m of isolated vertices. The size of the largest independent set of vertices in such a graph is easy to find. A maximum independent set contains one vertex from each of the E edges and it contains all m of the isolated vertices. Hence in this case, maxset = E + m = |V (G)| − |E(G)|, and we obtain a second try at a good algorithm in the following form. {maxset2} How much have we improved the complexity estimate?

Our next remark is that each value of i = 1, 2, . . , n is equally likely to occur. The reason for this is that we chose the splitter originally by choosing a random array entry. Since all orderings of the array entries are equally likely, the one that we happened to have chosen was just as likely to have been the largest entry as to have been the smallest, or the 17th -from-largest, or whatever. Since each value of i is equally likely, each i has probability 1/n of being chosen as the residence of the splitter.

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Algorithms and Complexity by wilf


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