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 B.4, Problem 1E
Program Plan Intro
To prove:
The total number of handshaking lemma in terms of undirected graph, can be defined as:
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In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E)
with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v)
such that 1< c(v)
Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is
the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is
the greatest distance between two of its vertices.
a) Show that if A(G) 24 then A(G) <2.
b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no
neighbors.
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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- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forward4. Let G=(V, E) be the following undirected graph: V = {1, 2, 3, 4, 5, 6, 7, 8} E = {(1, 7) (1, 4), (1, 6), (6, 4), (7, 5), (2, 5), (2, 4), (4, 3), (4, 8), (8, 3), (3, 5)}. a. Draw G. b. Is G connected? c. Give the adjacency matrix for the graph G given above. d. Determine whether the graph has a Hamiltonian cycle. If the graph does have a Hamiltonian cycle, specify the nodes of one. If it does not, prove that it does not have one. e. Determine whether the graph has a Euler cycle. If the graph does have a Euler cycle, specify the nodes of one. If it does not, prove that it does not have one.arrow_forwardConsider two graphs G and H. Graph G is a complete graph with 5 vertices; Graph H is a 2-regular graph with 10 vertices. How many edges are there in total between the two graphs?arrow_forwardGiven the following Graphs: Graph A: В 12c 1 4. E F G- H 4 J K 3 Graph B: Graph B is the undirected version of Graph A. 3.arrow_forward(V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the abovearrow_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_forward5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)arrow_forwardGiven an undirected graph G = (V, E), a vertex cover is a subset of V so that every edge in E has at least one endpoint in the vertex cover. The problem of finding a minimum vertex cover is to find a vertex cover of the smallest possible size. Formulate this problem as an integer linear programming problem.arrow_forwardQuestion 4 Given a undirected weighted graph G associate a number which is the weight of the heaviest edge in the tree. Find the spanning tree whose number is minimum among all spanning trees. Explain. (Hint: this involves a simple observation. The question is how to justify this observation. There are many ways to justify. The easiest is probably to recall where "exponentiation" appeared in the HW and use it.) (E,V). With each spanning tree wearrow_forwardPart 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…arrow_forwardLet V= {cities of Metro Manila} and E = {(x; y) | x and y are adjacent cities in Metro Manila.} (a) Draw the graph G defined by G = (V; E). You may use initials to name a vertex representing a city. (b) Apply the Four-Color Theorem to determine the chromatic number of the vertex coloring for G.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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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