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 35.2, Problem 1E
Program Plan Intro
To prove that
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Question 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.
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 .
2. A graph G = (V, E) satisfies |E| < 3 × |V| – 6. The
min-degree of G is defined as min,ev{degree(v)}. The
min-degree of G cannot be
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- 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.arrow_forwardLet 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.arrow_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
- Let G be a graph such that |V(G)| = |E(G)|. Show that δ(G) < 3.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_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_forward
- a. Show that if G is a simple graph with n vertices (wheren is a positive integer) and each vertex has degree greater than orequal to (n−1)/2, then the diameter of G is 2 or less. b. If G is a (not necessarily simple) graph with n verticeswhere each vertex has degree greater than or equal to (n−1)/2, is thediameter of G necessarily 2 or less? Either prove that the answer tothis question is "yes" or give a counterexample. Part a has already been answered so I only need help with part b. Part a is included for contextarrow_forward1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEVarrow_forwardA 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_forward
- Suppose 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_forwardLet G be a connected planar graph with 32 vertices, 50 edges and degree of each region is k. Find the value of k.arrow_forward2. Prove or disprove: For each positively weighted, connected, undirected graph G (with distinct weights), for each vertex v in G, the MST of G must contain the minimum-weight edge incident on v.arrow_forward
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