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- Given a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesfor the graphWhat edges are selected to obtain the minimum spanning tree? Select one or more than one:3. From the graph above determine the vertex sequence of the shortest path connecting the following pairs of vertex and give each length: a. V & W b. U & Y c. U & X d. S & V e. S & Z 4. For each pair of vertex in no. 3 give the vertex sequence of the longest path connecting them that repeat no edges. Is there a longest path connecting them?
- Is it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false.a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v. b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H.c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v? d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.A mathematician applies Prim’s Algorithm to find a minimum spanning tree for the weighted graph starting at vertex D. The order of the edges picked so far is AD, AB, BE, and EF. The next edge selected when applying Prim’s Algorithm should be BF FG CG CE What is the total weight of the tree that uses the edges DE, CE, CG, FG and BF? 93 92 91 90 The tree described in problem #8 is a spanning True FalseConsider an undirected graph on 8 vertices, with 12 edges given as shown below: Q2.1 Give the result of running Kruskal's algorithm on this edge sequence (specify the order in which the edges are selected). Q2.2 For the same graph, exhibit a cut that certifies that the edge ry is in the minimum spanning tree. Your answer should be in the form E(S, V/S) for some vertex set S. Specifically, you should find S.
- Draw out the Minimum Spanning Tree of the above graph.The following table presents the implementation of Dijkstra's algorithm on the evaluated graph G with 8 vertices. a) What do the marks (0, {a}) and (∞, {x}) in the 1st row of the table mean? b) What do the marks marked in blue in the table mean? c) Reconstruct all edges of the graph G resulting from the first 5 rows of the table of Dijkstra's algorithm. d) How many different shortest paths exist in the graph G between the vertices a and g?LetG= (V, E) be a graph. A vertex cover inGis a subset of verticesS, so thatfor every edge e= ab∈E, either a∈Sor b∈S. Now T= (V, E) be a tree with non-negative weights assigned to its vertices. Describe a linear time algorithm for computing a vertex cover of minimum weight. (Weight of a vertex cover is sum of vertex weightsinit.)You should define all necessary terms in your proposed solution, and clearly outline your algorithm.
- How to find the shortest paths in the following graph from node N to each of the other nodes using Dijksra’s Alogrithm?? I'm confused. Please help me.True or false: For graphs with negative weights, one workaround to be able to use Dijkstra’s algorithm (instead of Bellman-Ford) would be to simply make all edge weights positive; for example, if the most negative weight in a graph is -8, then we can simply add +8 to all weights, compute the shortest path, then decrease all weights by -8 to return to the original graph. Select one: True FalseLet 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?