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, Problem 3P
(a)
Program Plan Intro
To show that the graph contains Euler tour if and only if in-degree equals to out-degree.
(b)
Program Plan Intro
To determine the time complexity of the Euler tour
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The total degree of an undirected graph G = (V,E) is the sum of the degrees of all the vertices in V. Prove that if the total degree of G is even then V will contain an even number of vertices with uneven degrees. Handle the vertices with even degrees and the vertices with uneven degrees as members of two disjoint sets.
A Hamiltonian path on a directed graph G = (V, E) is a path that visits each vertex in V exactly once. Consider the following variants on Hamiltonian path:
(a) Give a polynomial-time algorithm to determine whether a directed graph G contains either a cycle or a Hamiltonian path (or both). Given a directed graph G, your algorithm should return true when a cycle or a Hamiltonian path or both and returns false otherwise. (b) Show that it is NP-hard to decide whether a directed graph G’ contains both a cycle and a Hamiltonian Path, by giving a reduction from the HAMILTONIAN PATH problem: given a graph G, decide whether it has a Hamiltonian path. (Recall that the HAMILTONIAN PATH problem is NP-complete.)
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
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