By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

**Read Online or Download Algorithms and Complexity (Second edition) PDF**

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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? 6. Graphs 47 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.

N − k)! k Some of their special values are: µ ¶ µ ¶ n n = 1 (∀n ≥ 0); = n (∀n ≥ 0); 0 1 µ ¶ µ ¶ n n = n(n − 1)/2 (∀n ≥ 0); = 1 (∀n ≥ 0). 2 n ¡ ¢ It is convenient to define nk to be 0 if k < 0 or if k > n ≥ 0. 11. For each ¡n¢ n ≥ 0, a set of n objects has exactly 2 subsets, and of these, exactly k have cardinality k ( ∀k = 0, 1, . . , n). There are exactly n! diﬀerent sequences that can be formed from a set of n distinct objects. Since every subset of [n] has some cardinality, it follows that: n µ ¶ X n = 2n (n = 0, 1, 2, .

Say that v and w are equivalent if there is a path of G that joins them. Let S be one of the equivalence classes of vertices of G under this relation. The subgraph of G that S induces is called a connected component of the graph G. A graph is connected if and only if it has exactly one connected component. , one in which vk = v1 . A cycle is a circuit if v1 is the only repeated vertex in it. We may say that a circuit is a simple cycle. We speak of Hamiltonian and Eulerian circuits of G as circuits of G that visit, respectively, every vertex, or every edge, of a graph G.

### Algorithms and Complexity (Second edition) by Herbert S. Wilf

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