Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Question
Chapter B, Problem 2P
a
Program Plan Intro
To reword each of the following statements as a theorem about undirected graphs and to prove them.
b
Program Plan Intro
To reword each of the following statements as a theorem about undirected graphs and prove them.
c.
Program Plan Intro
To reword each of the following statements as a theorem about undirected graphs and prove them.
d.
Program Plan Intro
To reword each of the following statements as a theorem about undirected graphs and prove them.
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Answer True or False to the following claims:
a. If G is graph on at most 5 vertices and every vertex has degree 2, then G is a cycle.
b. Let G be a forest. If we add an edge to G, then G is no longer a forest.
c. Let G be a graph, and let u, v, and w be vertices in G.
Suppose that a shortest path from u to v in G is of length 3, andsuppose that a shortest path from v to w in G is of length 4.
Then a shortest path from u to w in G is of length 7.
The total degree of an undirected graph G = (V,E) is the sum of the degrees of all the vertices in V. Prove that if the total degree of G is even then V will contain an even number of vertices with uneven degrees. Handle the vertices with even degrees and the vertices with uneven degrees as members of two disjoint sets.
Let G be a connected graph with n vertices and m edges.Which of the following statements are true?(i) G is a tree if and only if n = m+1.(ii) G is a tree if and only if m = n+1.(iii) G is a tree if and only if the addition of any edgeto G will produce a unique cycle.(iv) G is a tree if and only if G contains at least onepath b etween any two vertices.(v) If G is connected, then G is a tree if and only if Gcontains at most one path between any two vertices.(vi) A connected subgraph of a tree is always a tree.
Chapter B Solutions
Introduction to Algorithms
Ch. B.1 - Prob. 1ECh. B.1 - Prob. 2ECh. B.1 - Prob. 3ECh. B.1 - Prob. 4ECh. B.1 - Prob. 5ECh. B.1 - Prob. 6ECh. B.2 - Prob. 1ECh. B.2 - Prob. 2ECh. B.2 - Prob. 3ECh. B.2 - Prob. 4E
Ch. B.2 - Prob. 5ECh. B.3 - Prob. 1ECh. B.3 - Prob. 2ECh. B.3 - Prob. 3ECh. B.3 - Prob. 4ECh. B.4 - Prob. 1ECh. B.4 - Prob. 2ECh. B.4 - Prob. 3ECh. B.4 - Prob. 4ECh. B.4 - Prob. 5ECh. B.4 - Prob. 6ECh. B.5 - Prob. 1ECh. B.5 - Prob. 2ECh. B.5 - Prob. 3ECh. B.5 - Prob. 4ECh. B.5 - Prob. 5ECh. B.5 - Prob. 6ECh. B.5 - Prob. 7ECh. B - Prob. 1PCh. B - Prob. 2PCh. B - Prob. 3P
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- Q1) how many graphs are there on 20 nodes? (To make this question precise, we have to make sure we known what it mean that two graphs are the same . For the purpose of this exercise,we consider the nodes given and labeled,say,asAlice ,Bob,...... The graph consisting of a single edge connecting Alice and Bob is different from the graph consisting of a single edge connecting Eve and frank.) Q2) Formulate the following assertion as a theorem about graphs and prove it :At every party one can find two people who know the same number of other people (like Bob and Eve in our first example).arrow_forward5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…arrow_forwardThe problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy. ALso provide an expression using qualifiers. You need to provide a clear expression using the qualifiersarrow_forward
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