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
Program Plan Intro
To show that transpose of component graph GT is same as component graph G .
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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.
Given a spanning forest T of an undirected simple graph G = (V, E), the following hold.
Hall's theorem
Let d be a positive integer. We say that a graph is d-regular if every node has degree exactly d.
Show that every d-regular bipartite graph G = (L ∪ R, E) with bipartition classes L and R has |L| = |R|.
Show that every d-regular bipartite graph has a perfect matching by (directly) arguing that a minimum cut of the corresponding flow network has capacity |L|
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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- If a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.arrow_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_forwardSuppose 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_forward
- Prove that if G is a connected graph, then there always is a closed walk that passes through each edge at least once and at most twice.arrow_forwardGiven a directed graph G(V, E), find an O(|V | + |E|) algorithm that deletes at most half of its edges so that the resulting graph doesn’t contain any directed cycles. Also follow up with proof of correctness.arrow_forwardLet G be a graph such that |V(G)| = |E(G)|. Show that δ(G) < 3.arrow_forward
- Consider a subset S of the edges of an undirected graph Garrow_forwardLet 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_forward
- Prove 1 For a graph G = (V, E), a forest F is any set of edges of G that doesnot contain any cycles. M = (E, F) where F = {F ⊆ E : F is a forest of G} is amatroid.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_forwardLet G be a graph and R be a subset of its vertices. A subset of vertices DR is called R-dominating target if every connected subgraph of G containing Da dominates R. In addition, if each vertex of DR has some R verter in its closed neighborhood, then we call it an essential-R-dominating-target.arrow_forward
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