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 34.2, Problem 6E
Program Plan Intro
To show that there exists a Hamiltonian path from u to v in graph G which belongs to NP.
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. Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is Hamiltonian
Say 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.
Suppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- SUBJECT: GRAPH ALGORITHMS Prove that if v0 and v1 are distinct vertices of a graph G = (V,E) and a path exists in G from v0 to v1 , then there is a simple path in G from v0 to v1 .arrow_forwardA Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once. Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}. Give a graph G containing vertex set {fit, sit, cruise, cow, 2023, spinner, filler} and an edge set E such that there is a Hamiltonian path, where "cruise" appears according to its alphabet order in the path;arrow_forwardDraw a simple, connected, weighted graph with 8 vertices and 16 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph. Problem R-14.23 in the photoarrow_forward
- 1. Prove that if v1 and v2 are distinct vertices of a graph G = (V,E) and a path exists in G from v1 to v2 , then there is a simple path in G from v1 to v2 .arrow_forwardA Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once. Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}. Prove that Ham-Path belongs to NP, i.e. given any graph G and a pair of nodes u and v and apath p between u and v, there is a polynomial algorithm to check whether p is in Ham-Patharrow_forwardA Hamiltonian cycle of an undirected graph G = (V, E) is a simple cycle that contains each vertex in Vonly once. A Hamiltonian path of an undirected graph G = (V, E) is a simple path that contains each vertexin V only once. Let Ham-Cycle = {<G, u, v> : there is a Hamiltonian cycle between u and v in graph G} andHam-Path = {< G, u, v>: there is a Hamiltonian path between u and v in graph G}. Prove that Ham-Path is NP-hard using the reduction technique, i.e. find a known NP-completeproblem L ≤p Ham-Path (reduce from the Ham-Cycle problem). Make sure to give the followingdetails: 1) Describe an algorithm to compute a function f mapping every instance of L to an instance of Ham-Path 2) Prove that x ∈ L if and only if f(x) ∈ Ham-Path 3) Show that f runs in polynomial timearrow_forward
- We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…arrow_forwardINSTANCE: A plane graph G = (V, E) (that is a graph embedded in the plane without intersections), and two of its vertices, and t. QUESTION: Does G contain a Hamiltonian path from to t.arrow_forwardProve that in any simple graph G (i.e. no multiple edges, no loops) there exist two distinct vertices with the same degrees.arrow_forward
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