Q2. Find a maximal matching M₁ in the graph below G such that which is not a maximum matching M2. Then obtain the symmetric differences of the M₁ and M2 (M₁ A M₂). a x c²
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- Let G = (U, W, E; l) be an edge-weighted bipartite graph with l (e) > 0 each e E E. Suppose that the total weight of the edges incident to each vertex in UUW is exactly one. Show that G has a perfect matching. (Hint: Use the Hall's condition.)ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.The total degree of an undirected graph G = (V,E) is the sum of the degrees of all the vertices in V. Prove that if the total degree of G is even then V will contain an even number of vertices with uneven degrees. Handle the vertices with even degrees and the vertices with uneven degrees as members of two disjoint sets.
- Suppose we have the following undirected graph, and we know that the two bolded edges (B-E and G-E) constitute the global minimum cut of the graph. 1. If we run the Karger’s algorithm for just one time to find the global minimum cut, what is the probability for the algorithm to find the minimum cut correctly? Please show your reasoning process. 2. How many times do we need to run the Karger’s algorithm if we want to guarantee that the probability of success is greater than or equal to 0.95, by “success” we mean that there is at least one time the Karger’s algorithm correctly found the minimum cut. Please show your reasoning process. [You do not have to work out the exact value of a logarithm]Let G = (V,E) be a bipartite graph, but this time it is a weighted graph. The weight of acomplete matching is the sum of the weights of its edges. We are interested in finding aminimum-weight complete matching in G.a) Give a legitimate C for a branch-and-bound (B&B) algorithm that finds a minimum-weightcomplete matching in G, and prove that your C is valid. Your C cannot be just the cost sofar.b) Using your C, apply B&B to find a minimum-weight complete matching in the followingweighted bipartite graph G: A = (1,2,3), B = (4,5,6,7)E = (((1,4), 3]. [(1,5), 4]. [(1,7). 15]. ((2,4), 1]. |(2,5), 8]. [(2,6), 3]. |(3,4), 3]. [(3,5), 9]. [ (3,6). 51).Show the solution tree. the C of every tree node generated, and the optimal solution. Also,mark the order in which each node in the solution tree is visited.Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.
- Let G be a graph with n vertices representing a set of gamers. There is an edge between two nodes if the corresponding gamers are friends. You want to partition the gamers into two disjoint groups such that no two gamers in the same group are friends. Is the problem in P or in NP? Give a formal proof for your answer. A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.Given N cities represented as vertices V₁, V2,..., UN on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices v; and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). Assume that all edges have different values, and e₁, has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.Consider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.
- Consider a graph G, where each of the edges have different weights. Let T1 be the minimum-weight spanning tree produced by Kruskal's Algorithm, and let T2 be the minimum-weight spanning tree produced by Prim's Algorithm. I claim that T1 and T2 must be identical spanning trees - i.e., the exact same set of edges must appear in both trees. Determine whether this claim is TRUE or FALSE. If your answer is TRUE, see if you can figure out why the claim is true. If your answer is FALSE, see if you can come up with a counterexample.You are given a bipartite graph G=(U,V,E), and an integer K. U and V are the two bipartitions of the graph such that |U| = |V| = N , and E is the edge set. The vertices of U are {1,2,...,N } and that of V are {N+1,N+2,...,2N }. You need to find out whether the total number of different perfect matchings in G is strictly greater than K or not. Recall that a perfect matching is a subset of E such that every vertex of the graph belongs to exactly one edge in the subset. Two perfect matchings are considered to be different even if one edge is different. Write a program in C++ programming language that prints a single line containing “Perfect” if the number of perfect matchings is greater than K, and “Not perfect” in other cases.Sample Input:3 5 21 42 62 53 53 5Output:Not PerfectQ1a Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k. 1b INDEPENDENT SET has the following Decision Problem: given a graph G and an integer k₁, does G have an independent set of size at least k₁? VERTEX COVER has the following Decision Problem: given a graph G and an integer k2, does G have a vertex cover of size at most k2? It is known that INDEPENDENT SET is NP-complete. Prove that INDEPENDENT SETSEE MORE QUESTIONS
