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R-13.6 - Suppose we represent a graph G having n vertices and m edges with the edge list structure. Why, in this case, does the insertVertex function run in O(1) time while the eraseVertex function runs in O(m) time?
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- Your task for this assignment is to identify a spanning tree in a connected undirected weighted graph using C++. 1. 2. 3. 4. Implement a spanning tree algorithm using C++. A spanning tree is an acyclic spanning subgraph of the of a connected undirected weighted graph. Your program will be interactive. Graph edges with respective weights (i.e., v1 v2 w) are entered at the command line and results are displayed on the console. Each input transaction represents an undirected edge of a connected weighted graph. The edge consists of two unequal uppercase letters representing graph vertices that the edge connects. Each edge has an assigned weight. The edge weight is a positive integer in the range 1 to 99. The three values on each input transaction are separated by a single space. An input transaction containing the string “end-of-file" signals the end of the graph edge input. After the edge information is read, the spanning tree evaluation process begins. Use an adjacency matrix for…the number of edges in a complete 2 of 2 undirected graph of 80 vertices is.? Select one: a. exactly equals 3160 b. None c. less than or equals 3160 d. less than or equals 6320 e. exactly equals 6320 Assume that you have a doubly link list. Pointer x is pointing to the last node in the link list and pointer y is pointing to before the last node. which of the following will remove the node pointed by x from the list? Select one: O a. delete x; b. y->prev->next = y->next; X->prev= y->prev; delete x; O C. X->prev->next = x->next; y->prev= x->prev; delete x; O d. x->next = y; y->prev = x->prev; delete x;5. 6. Given an undirected graph with n vertices and m edges, find an O(n+m) time algorithm that determines whether it is possible to color all the vertices red and blue such that every edge is between a red vertex and blue vertex. If such a coloring exists, your algorithm should produce one. Give an algorithm that determines whether or not a given undirected graph contains a cycle. Your algorithm should run in time O(n) and independent of m.
- Define a function breadthFirst, which performs a breadth-first traversal on a graph, given a start vertex. This function should return a list of the labels of the vertices in the order in which they are visited. Test the function thoroughly with the case study program.(In Python with comments) Use this template: class Vertex: def __init__(self, label): self.label = label self.neighbors = [] def __str__(self): return self.label def add_neighbor(self, neighbor): self.neighbors.append(neighbor) self.neighbors.sort() # Graph definition provided, please don't modify any of the provided code# Add any helper methods if necessaryclass Graph: def __init__(self): self.vertices = [] # Add a vertex object def add_vertex(self, vertex): self.vertices.append(vertex) # Get index of a vertex with corresponding label # Return -1 if not found def get_vertex_index(self, label): index = 0 for vertex in…Consider a linked list (SLL or DLL your choice) in which, to delete anelement from the list, we simply mark that node for the deletion.Whenever the number of deleted nodes in the list exceeds the number ofnon-deleted nodes, we simply delete all nodes which were marked fordeletion. Implement the following operations for such a linked list.1. addFirst2. addLast3. delFirst4. delLast5. Delete (mark for deletion) an ith node in the list.6. display all non-deleted nodes in the list7. Size : number of non-deleted nodes in the list.8. Recall the last deleted node in the list.Given a singly linked list L, where x and y are two data elements that occupy the nodes NODEX and NODEY with PREVIOUSX as the node, which is the previous node of NODEX, write a pseudo-code to swap the date x and y in list L by manipulating the links only (data swapping is not allowed). Assume that x and y are available in the list and are neither neighbors nor the end nodes of list L. For example, given the list L shown in Figure P6.10(a), with L, NODEX, NODEY and PREVIOUSX marked on it, the swapping should yield the list shown in Figure P6.10(b). NODEX and NODEY are neither immediate neighbors nor the end nodes of list L. PREVIOUS X NODE X PREVIOUS X எக்க்கக்கா எத்த்தக்க (a) Before swapping g and x NODE Y NODE Y (b) After swapping g and x Linked Lists 193 NODE X Figure P6.10. Swapping of elements in a singly linked list by manipulating links
- Given a singly linked list L, where x and y are two data elements that occupy the nodes NODEX and NODEY with PREVIOUSX as the node, which is the previous node of NODEX, write a pseudo-code to swap the date x and y in list L by manipulating the links only (data swapping is not allowed). Assume that x and y are available in the list and are neither neighbors nor the end nodes of list L. For example, given the list L shown in Figure P6.10(a), with L, NODEX, NODEY and PREVIOUSX marked on it, the swapping should yield the list shown in Figure P6.10(b). NODEX and NODEY are neither immediate neighbors nor the end nodes of list L. PREVIOUS X NODE Y 00 bodo a X С NODE X PREVIOUS X g W (a) Before swapping g and x NODE Y Linked Lists 193 NODE X (b) After swapping g and x addada. Figure P6.10. Swapping of elements in a singly linked list by manipulating linksGiven a linked list of N nodes such that it may contain a loop. A loop here means that the last node of the link list is connected to the node at position X(1-based index). If the link list does not have any loop, X=0. Remove the loop from the linked list, if it is present, i.e. unlink the last node which is forming the loop. Example 1: Input: N = 3 value[] = {1,3,4} X = 2 Output: 1 Explanation: The link li. st looks like 1 -> 3 -> 4 ^ | |____|10. Write a C program to find maximum matching in bipartite graph. A graph is said to be bipartite if all its vertices can be partitioned into two disjoint subsets V and U so that every edge connects a vertex in V with a vertex in U. The following graph is a bipartite graph. The adjacency matrix of the input bipartite graph is the following. 1 1 0 0 1 0 0 0 10 10 0 0 1 Lo o o 1 1. 1 1. The maximum matching for the graph will be The solution is represented as ro 1 0 0 10 0 0 0 0 0 10 0 0. 1. 0 0 1-
- Here is a program that creates the minimum spanning tree for the graph inFigure 16.7:Undirected graph is given with the list of edges. First line contains number of vertices. Each next line contains an edge. Run Depth First Search (DFS) from the specified vertex. Print the last visited vertex during DFS. Input The first line contains number of vertices n. Next lines contains list of edges (read till the end of file). Output Run Depth First Search (DFS) from the vertex 5. Print the last visited vertex during DFS. Sample input 5 13 23 25 Sample output 1This task is solved in Python. 3. Write a function build_my_graph2 () that: a) creates the following Graph. b) runs Depth First Search (DFS) algorithm starting from node 'a' and prints all the visited nodes. What is printed in the output when you run the function? Note: you can use the implementation of Graph class and the DFS algorithm (provided in Lecture notes). 5.0 g h e