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 1E
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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.
let a graph have vertices h,i,j,k,l,m,n,o and edge set {{h,i},{h,j},{h,k},{h,n},{h,o},{j,k},{j,l},{j,m},{k,l},{m,o},.
on paper, draw the graph. then answer the questions below.
a. what is the degree of vertex k?
b. what is the degree of vertex h?
c. how many connected components does the graph have?
Let G be a directed graph where each edge is colored either red, white, or blue. A walk in G is called a patriotic walk if its sequence of edge colors is red, white, blue, red, white, blue, and soon. Formally ,a walk v0 →v1 →...vk is a patriotic walk if for all
0≤i<k, the edge vi →vi+1 is red if i mod3=0, white if i mod3=1,and blue if i mod3=2.
Given a graph G, you wish to find all vertices in G that can be reached from a given vertex v by a patriotic walk. Show that this can be accomplished by efficiently constructing a new graph G′ from G, such that the answer is determined by a single call to DFS in G′. Do not forget to analyze your algorithm.
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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- Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).arrow_forwardAn edge is called a bridge if the removal of the edge increases the number of connected components in G by one. The removal of a bridge thus separates a component of G into two separate components. Let G be a graph on 6 vertices and 8 edges. How many bridges can G have at the most? (The answer is an integer)arrow_forwardHow many edges does a graph have if its degree sequence is 2, 4, 4, 5, 3?A. Draw a graph with the above listed sequence.B. Is it possible to draw an Euler Circuit with such a sequence of vertex degrees?Is it possible to draw an Euler Path? If yes, to either of these questions, draw the a graph that supports your answer.arrow_forward
- Consider the graph depicted below. Answer the following questions: What is the degree of vertex a? What is the sum of degrees of all vertices? Who are the neighbors of vertex f? [Please, give the list in alphabetical order, separating letters with only a comma. Do not insert blank spaces] How many zeros does the adjacency matrix of the graph above contain? What is the value on row g and column d of the adjacency matrix?arrow_forwardRun a Program to experiments to determine empirically the probability that DepthFirstPaths finds a path between two randomly chosen vertices and to calculate the average length of the paths found, for various graph models.arrow_forwardWe have a Directed Weighted Graph with positive edge weights. Let us think the current shortest path in the graph is p to q. Suppose we change each edge weight in the graph by taking cube root of each weight. Will the shortest path remain the same or will it change for the new graph? Give an argument for or counterexamples for this. Could someone please help me answer this with explanation and examples along with a graph diagram.Thank youarrow_forward
- Let G be a graph with 7 vertices, where eachvertex is labelled by a number from 1 to 7. Twovertices are selected at random. Let us call themu and v. Now, directed edges are drawn from u toall other vertices except v and directed edges aredrawn from all vertices to v except from u. Let xbe the total possible topological sortings of G.hint: You may remember the terms factorials orfibonacci?Write down the value of x:arrow_forwardRun experiments to determine empirically the probability that BreadthFirstPaths finds a path between two randomly chosen vertices and to calculate the average length of the paths found, for various graph models.arrow_forwardCan you help me with this problem and can you do it step by step. question: Show that every graph with two or more nodes contains two nodes that have equal degrees.arrow_forward
- Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v]. 3. Write the full pseudocode for the following problem.Input: A directed graph G, and an ordering on the vertices given in a list A.Output: Is A a topological order? In other words, is there an i, j such that i < j and there is an edge fromA[j] to A[i]?arrow_forwardConnecting a Graph, Let G be an undirected graph. Give a linear time algorithm to compute the smallest number of edges that one would need to add to G in order to make it a connected graph??arrow_forwardIs it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false.a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v. b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H.c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v? d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.arrow_forward
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