Chapter 11: Q1E (page 795)
How many edges must be removed from a connected graph with n vertices and m edges to produce a spanning tree?
The edges are=m-n+1.
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a. Describe Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees.
b. Illustrate how Kruskal's algorithm and Prim's algorithm are used to find a minimum spanning tree, using a weighted graph with at least eight vertices and \(15\) edges.
a) How many edges does a tree with \({\bf{n}}\) vertices have?
b) What do you need to know to determine the number of edges in a forest with \({\bf{n}}\) vertices?
Show that every tree can be colored using two colors. The rooted Fibonacci trees \({\bf{Tn}}\) are defined recursively in the following way. \({\bf{T1}}\)and\({\bf{T}}2\) are both the rooted tree consisting of a single vertex, and for \({\bf{n = 3, 4,}}...{\bf{,}}\) the rooted tree \({\bf{Tn}}\) is constructed from a root with \({\bf{Tn - }}1\) as its left subtree and \({\bf{Tn - 2}}\) as its right subtree.
Answer these questions about the rooted tree illustrated.
Show that if no two edges in a weighted graph have the same weight, then the edge with least weight incident to a vertex v is included in every minimum spanning tree.
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