The minimum spanning tree of an undirected graph G exists if and only if G is connected. True or False?
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The minimum spanning tree of an undirected graph G exists if and only if G is...
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is increased. The input to your algorithm should be the edge e and its new weight: your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is decreased. The input to your algorithm should be the edge e and its new weight; your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
C++ programing question22
Minimum spanning tree
Time limit: 1 second
Problem Description
For a connected undirected graph G = (V, E), edge e corresponds to
a weight w, a minimum weight spaning tree can be found on the
graph.
Into trees.
Input file format
At the beginning, there will be a positive integer T, which means
that there will be T input data.
The first line of each input has two positive integers n,m,
representing n points and m edges...
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
Problem 3: Bounded-Degree Spanning Trees (10 points). Recall the minimum spanning tree problem studied in class. We define a variant of the problem in which we are no longer concerned with the total cost of the spanning tree, but rather with the maximum degree of any vertex in the tree. Formally, given an undirected graph G = (V,E) and T ⊆ E, we say T is a k-degree spanning tree of G if T is a spanning tree of G,...
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
Answer the following true or false questions with a brief justification. A) There exists an undirected graph on 6 vertices whose degrees are 4, 5, 8, 9, 3, 6. B) Every undirected graph with n vertices and n − 1 edges is a tree. C) Let G be an undirected graph. Suppose u and v are the only vertices of odd degree in G. Then G contains a u-v path.
Use Kruskals Algorithm to find the minimum spanning tree for the weighted graph. Give the total weight of the minimum spanning tree. What is the total weight of the minimum spanning tree? The total weight is _______
P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are no edges between nodes at the same level in any BFS tree for G. (An undirected graph is defined to be bipartite if its nodes can be divided into two sets X and Y such that all edges have one endpoint in X and the other in Y.)
P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are...
3. A Unicvcle Problem Prove that a cycle exists in an undirected graph if and only if a BFS of that graph has a cross-edge. (**) Your proof may use the following facts from graph theory . There exists a unique path between any two vertices of a tree. . Adding any edge to a tree creates a unique cycle.