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 23.1, Problem 3E
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To show that if an edge
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Show that the smallest edge must be present in a graph component's minimal spanning tree.
Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.
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.
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- Be 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_forwardGiven a complete graph with 5 nodes, and different weights of the edges, how do you find the minimum spanning tree using Prim's method?arrow_forwardYou are given a simple connected undirected graph with N nodes(or vertices) for computing a Minimum Spanning Tree (MST) of the graph. Write down the upper and lower bound of the possible number of edges of that MST. Now consider the same for a “simple disconnected undirected graph”. What is the lower and upper bound now?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_forwardDraw a connected graph with 7 nodes and 10 edges which has no cutnode nor bridge. Find by the use of reduction algorithm the number of spanning trees in this graph.arrow_forwardLet the graph G be a cycle of n nodes in which x edges have the weight 100 and y edges have weight 200. How many minimum spanning trees does G have?arrow_forward
- Prove that if an edge (u, v) is in a MST for a graph G, then (u, v) is a light edge crossing some cut in G.arrow_forwardProve that a minimum spanning tree of a graph component must include a shortest edgearrow_forwardShow that a connected graph is a tree if and only if every edge is a cut edge.arrow_forward
- Let G be a directed acyclic graph with exactly one source r such that for any other vertex v there exists a unique directed path from r to v. Let Gu be the undirected graph obtained by erasing the direction on each edge of G. Prove that (Gu,r) is a rooted tree.arrow_forwardShow that every graph with two or more nodes contains two nodes that have equal degreearrow_forwardLet G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setarrow_forward
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