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 22.5, Problem 4E
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To show that transpose of component graph GT is same as component graph G .
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Problem 4. Let G = (V,E) be an undirected connected graph with maximum edge weight Wmax. Provethat if an edge with weight Wmax appears in some MST of G, then all MSTs of G contain an edge withweight Wmax.
Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.
Show that if all edges of a graph G have pairwise distinct weights, then thereis exactly one MST for G.
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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- A set of vertices in a graph G = (V,E) is independent if no two of them are adjacent. Let G = (V,E) be an undirected graph with subset I of V an independent set. Let the degree of each vertex in V be at least 2. Also let |E| - E a ɛl deg(a) + 2 ||| < |V| Can G have a Hamiltonian cycle?arrow_forwardLet G = (V, E) be a connected graph with a positive length function w. Then (V, d) is a finite metric space, where the distance function d is defined asarrow_forwardLet G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.arrow_forward
- If a graph has a collection of subsets of the edge set E, with the edges of at most one cycle: Show that if |X| and |Y| are independent sets, show that |X| < |Y| implies that there exists {m} E Y\X such that X U {m} is independent.arrow_forwardSuppose we have a graph G = (V, E) with m edges. Prove that there exists a partition of V into three subsets A, B, C such that there are 2m edges between these subsets (i.e. between A and B, between B and C, or between A and C). 3arrow_forwardGiven the following adjacency matrix of the graph G. Is the graph is connected, regular, and complete? [0 1 1 1] 1 0 1 0 1 10 1 li 0 1 0] G=arrow_forward
- 1. Consider the following directed graph. A F J D G В E H I 1. What are the sources and sinks of the graph? 2. Give one linearization of this graph. 3. How many linearization does this graph have?arrow_forwardLet G be a graph such that |V(G)| = |E(G)|. Show that δ(G) < 3.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_forward
- Let G be a graph with a connected subgraph H. Prove that H is a subgraph of a unique connected component of G.arrow_forwardShow that an MST of an undirected graph is equivalent to abottleneck SPT of the graph: For every pair of vertices v and w, it gives the path connecting them whose longest edge is as short as possible.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_forward
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