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 22.3, Problem 6E
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To show that in an undirected graph classifying the edge encountered first is equivalent to classifying it according to the ordering in classification scheme in DFS.
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If 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.
Every set of vertices in a graph is biconnected if they are connected by two disjoint paths. In a connected graph, an articulation point is a vertex that would disconnect the graph if it (and its neighbouring edges) were removed. Demonstrate that any graph that lacks articulation points is biconnected. Given two vertices s and t and a path connecting them, use the knowledge that none of the vertices on the path are articulation points to create two disjoint paths connecting s and t.
An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex.
NOTE: graphs are in the image attached.
Which of the graphs below have Euler paths? Which have Euler circuits?
List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits?
Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path?
What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs.
Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? NOTE: graphs is in the image attached.
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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- Using EXACTLY three nodes and three edges per graph, draw thefollowing graphs: (a) unweighted and undirected, (b) a DAG, (c) directed and connected, and (d)weighted, directed, and disconnected.arrow_forwardComputer Science Frequently, a planar graph G=(V,E) is represented in the edgelist form, which for each vertex vi V contains the list of its incident edges, arranged in the order in which they appear as one proceeds counterclockwise around i v . Show that the edge-list representation of G can be transformed to the DCEL (DoublyConnected-Edge-List) representation in time O(|V|).arrow_forwardEvery set of vertices in a graph is biconnected if they are connected by two disjoint paths. In a connected graph, an articulation point is a vertex that would disconnect the graph if it (and its neighbouring edges) were removed. Demonstrate that any graph that lacks articulation points is biconnected. Hint: Given a set of vertices s and t and a path connecting them, take advantage of the fact that none of the vertices on the path are articulation points to create two disjoint paths connecting s and t.arrow_forward
- In breadth-first and depth-first search, an undiscovered node is marked discovered when it is first encountered, and marked processed when it has been completely searched. At any given moment, several nodes might be simultaneously in the discovered state. (a) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a breadth-first search starting from v. (b) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a depth-first search starting from v.arrow_forwardA 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_forwardShow that in an undirected graph, classifying an edge .u; / as a tree edge or a back edge according to whether .u; / or .; u/ is encountered first during the depth-first search is equivalent to classifying it according to the ordering of the four types in the classification scheme.arrow_forward
- 4. Run the Bellman-Ford algorithm on the directed graph given blow, using vertex z as the source. In each pass, relax edges in the order (t, x.), (t, y), (t, z), (x, t), (y, x), (y, z), (s, t), (s, y), (z, s), (z, x). Show the d and values after each pass. -2 8 y INarrow_forwardGive a circumstance in which an undirected graph does not contain an Eulerian cycle that is both adequate and optional. Explain your response.arrow_forward2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).arrow_forward
- A directed graph G= (V,E) consists of a set of vertices V, and a set of edges E such that each element e in E is an ordered pair (u,v), denoting an edge directed from u to v. In a directed graph, a directed cycle of length three is a triple of vertices (x,y,z) such that each of (x,y) (y,z) and (z,x) is an edge in E. Write a Mapreduce algorithm whose input is a directed graph presented as a list of edges (on a file in HDFS), and whose output is the list of all directed cycles of length three in G. Write the pseudocode for the mappers/reducers methods. Also, assuming that there are M mappers, R reducers, m edges and n vertices -- analyze the (upper-bound of the) communication cost(s).arrow_forwardSay that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.arrow_forward3. For the graph G= (V, E), find V, En all parallel edges, all loops and all isolated vertices and state whether G is a simple graph. Also state on which vertices edges es is incident and on which edges vertex v2 is incident. e1 U4arrow_forward
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