Elon Musk is running the graph construction business. A client has asked for a special graph. A graph is called special if it satisfies the following properties: It has <10' vertices. It is a simple, undirected, connected 3-regular graph. It has exactly k bridge edges, (k given as input). For a graph G, define f(G) to be the minimum number of edges to be removed from it to make it bipartite. The client doesn't like graphs with a high value of f, so you have to minimize it.If there doesn't exist any special graph, print-1. Otherwise, find a special graph G with the minimum possible value of f(G) and also find a subset of its edges of size f(G) whose removal makes it bipartite. In case there are multiple such graphs, you can output any of those.Elon has assigned this task to you, now you have to develop a C++ code that takes bridge edges as input and print all the possible graphs.

w describe the Python code for the 2-approximation algorithm. The main part of the code deals with deleting the randomly selected edge from the graph along with its neighboring edges. Input to the procedure VC_Approx is the incidence matrix B(G) of the graph G. The unmarked edges are held in the list edges and the endpoints of a selected edge are included in the list V C at each iteration. The incidence matrix B of the graph is input and we find adjacent edges to the matched edge by iterating through rows of B. Each found edge is deleted from the list edges and main while loop iterates until this list becomes empty. We check adjacent edges to the matched edge from this matrix by deleting adjacent edges at each endpoint.

Please help me with this practice problem in python : Implement two-level iterative method B = B_{TL} for graph Laplacian matrices. We want the symmetric B. Components: Given a graph, construct its graph Laplacian matrix. Then using Luby's algorithm, construct the P matrix that ensures a prescribed coarsening factor, e.g., 2, 4, or 8 times smaller number of coarse vertices. Since the graph Laplacian matrix is singular (it has the constants in its nullspace), to make it invertible, make its last row and columns zero, but keep the diagonal as it were (nonzero). The resulting modified graph Laplacian matrix A is invertible and s.p.d.. Form the coarse matrix A_c = P^TAP. To implement symmetric two-level cycle use one of the following M and M^T: (i) M is forward Gauss-Seidel, M^T - backward Gauss-Seidel (both corresponding to A) (ii) M = M^T - the ell_1 smoother. Compare the performance (convergence properties in terms of number of iterations) of B w.r.t. just using the smoother M…

3. 4. Given a directed acyclic graph G = (V, E) and two vertices s, te V, design an efficient algorithm that computes the number of different directed paths from s to t. Define the incidence matrix B of a directed graph with no self-loop to be an nxm matrix with rows indexed by vertices, column indexed by edges such that Bij = -1 1 0 if edge j leaves vertex i, if edge jenters vertex i, otherwise. Let BT be the transpose of matrix B. Find out what the entries of the n x n matrix BBT stand for.