Chapter 11: Q35E (page 796)
Explain how to use breadth-first search to find the length of a shortest path between two vertices in an undirected graph.
By using the breadth-first search it can get that the length of the shortest path between two vertices is an undirected graph.
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Show that Sollin’s algorithm requires at most \({\bf{logn}}\) iterations to produce a minimum spanning tree from a connected undirected weighted graph with \({\bf{n}}\) vertices.
a. Explain how backtracking can be used to determine whether a simple graph can be colored using \(n\) colors.
b. Show, with an example, how backtracking can be used to show that a graph with a chromatic number equal to \({\bf{4}}\) cannot be colored with three colors, but can be colored with four colors.
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.
Suppose that \({\bf{e}}\) is an edge in a weighted graph that is incident to a vertex v such that the weight of \({\bf{e}}\) does not exceed the weight of any other edge incident to v. Show that there exists a minimum spanning tree containing this edge.
Show that if \(G\) is a weighted graph with distinct edgeweights, then for every simple circuit of \(G\), the edge of maximum weight in this circuit does not belong to anyminimum spanning tree of \(G\).
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