Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 23.1, Problem 6E
Program Plan Intro
To show that a graph has a unique minimum spanning tree for every cut of the graph and also show that vice versa is not true by using an example.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A (k, l)-dumbbell graph is obtained by taking a complete graph on k (labeled) nodes and a complete graph on l (labeled) nodes, and connecting them by a single edge. Find the number of spanning trees of a dumbbell graph.
(a) Prove that for all n > 6, if a connected n-vertex graph has n+ 2 edges or fewer, then it
is planar.
(b) Prove that for all n> 6, there is a connected n-vertex graph with n +3 edges which is
not planar.
Assume that the Hamiltonian cycle problem for undirected graphs is NP-complete. Show that the Hamiltonian cycle problem is NP-complete for directed graphs.
Note: Just need to reduce the Hamiltonian cycle problem for an undirected graph to a directed graph: for any edge (u, v) in the undirected graph, add two directed edges u->v and v->u.
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- Please give me correct solution.arrow_forwardEnd of Solution In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G isarrow_forwardBe 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.arrow_forward
- Please give an explanation as to why your answer is correct.arrow_forwardQuestion 2: Let G be a planar graph on at least 4 vertices. Suppose that the minimum degree of G is 5. Show that the number of vertices of degree 5 in G is at least 12.arrow_forwardConsider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.arrow_forward
- Show that every nonincreasing sequence of nonnegative integers with an even sum of its terms is the degree sequence of a pseudograph, that is, an undirected graph where loops are allowed. [Hint: Construct such a graph by first adding as many loops as possible at each vertex. Then add additional edges connecting vertices of odd degree. Explain why this construction works.]arrow_forwardI need the answer as soon as possiblearrow_forwardConsider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle of length k (3 <= k <= n). Prove that the graph G must contain at least k vertices of degree 2.arrow_forward
- Suppose G = (V, E) is an undirected connected weighted graph such that all its edge weightsare distinct. Prove that that the minimum spanning tree of G is unique.arrow_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_forwardPart a) only pleasearrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill Education
Starting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSON
Digital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSON
Database Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage Learning
Programmable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education