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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Chapter 22.5, Problem 2E
Program Plan Intro
To show that the strongly connected component works on the graph G having vertices and adjacent vertices list in alphabetical order.
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Show the result of running BFS on the following undirected graph, usingvertex u as the source, assume that the vertices are considered in an alphabetical order. (Show thedistance, and list the order of vertices entering the Queue.)
Give an example of a connected, weighted, undirected graph such that neitherthe BFS-tree nor the DFS-tree is an MST, regardless of how the adjacency lists are orderedand at which vertex BFS or DFS are started. Argue that your graph satisfies the aboverequirements.
Let G be a connected graph, and let T1, T2 be two spanning trees. Prove thatT1 can be transformed to T2 by a sequence of intermediate trees, each obtainedby deleting an edge from the previous tree and adding another.
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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- Suppose We do a DFS on a directed graph Gd and G is corresponding depths first tree/forrest. if we remove from G all the back edges with respect to Gd the resulting graph will have no cycles. true or false?arrow_forwardSuppose G is a connected undirected graph. An edge e whose removal disconnects the graph is called a bridge. Must every bridge e be an edge in a depth-first search tree of G? Give a proof or a counterexample.arrow_forwardFor any given connected graph, G, if many different spanning trees can be obtained, is there any method or condition setting that allows the DFS spanning tree of G to only produce a unique appearance? can you give me some simple opinion?arrow_forward
- Suppose we have a topological sorting of the nodes in a directed acyclic graph (DAG). If a node appears before another node vvin the topological sort, then there must exist a path from u to v in the DAG. If true, explain your reasoning. If false, show a counterexample graph with a maximum of 5 nodes.arrow_forwardSuppose that G is an unconnected graph that consists of 4 connected components. The first component is K4, the second is K2,2, the third is C4 and the fourth is a single vertex. Your job is to show how to add edges to G so that the graph has an Euler tour. Justify that your solution is the minimum number of edges added.arrow_forwardlet us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proofarrow_forward
- A network is considered to be biconnected if every pair of its vertices is linked by two distinct paths. A vertex that, if it and its surrounding edges were removed, would result in the graph becoming unconnected is known as an articulation point in a linked network. show any graph without articulation points that it is biconnected. Use the fact that none of the vertices on the path is an articulation point to construct two disjoint paths connecting s and t given a set of vertices s and t and a path connecting them.arrow_forwardSuppose we do a BFS on a connected undirected graph G and T is corresponding depth-first tree. if we remove from G all the cross edges with respecr to T the result will just be a tree. True or False?arrow_forwardThe traversed graph is only acyclic if DFS doesn't find any back edges.arrow_forward
- How can I prove that G/e is connected? (If G is a connected graph and e is an edge of G) And what does a spanning tree has to do with this proof?arrow_forward. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.arrow_forwardIf a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.arrow_forward
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