Enumerate the number of perfect matchings in each of the following. Prove your answers. (а) Кп,я (b) K2n (c) The graph G = (X UY, E), where X = {x1,... , Tn} and Y : disjoint sets, and E := {x;Y; : i # j}. For example, if n = 4, then G is as follows: {yı,..., Yn} are %3D Y2 13 Y3 X Y
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- Please Answer this in Python language: You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj. For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁. You want to minimise maai[1..N) Di - mini[1..N) Di. Output the minimum possible value of this expression for a valid assignment as described above, and also print the corresponding assignment. Note: The given graph need not be connected. • If there are multiple possible assignments, output anyone. • Since the input is large, prefer using fast input-output methods. Input 1 57 12 13 14 23 24 25 35 Output 2 43251 QLet G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matchingIn the figure below there is a weighted graph, dots represent vertices, links represent edges, and numbers represent edge weights. S 2 1 2 1 2 3 T 1 1 2 4 (a) Find the shortest path from vertex S to vertex T, i.e., the path of minimum weight between S and T. (b) Find the minimum subgraph (set of edges) that connects all vertices in the graph and has the smallest total weight (sum of edge weights). 2. 3.
- TRY YOUR BEST PLEASE Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1. Starting with no edges between A and B, if |E| many edges are added between A and B uniformly at random, what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a potential perfect matching, let X sub S = 1 if all the edges in S are added to the graph, and X sub S= 0 if any of them are missing. What isE[X sub S]?3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)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 Perfect3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 15. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups X and Y, and each edge has one end point in X and one end point in Y. A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions: 1. Every vertex in X belongs to at most two edges in S. 2. Every vertex in Y belongs to at most one edge in S. Please provide an algorithm to find the size (number of edges) of maximum 2-1 generalized matching.The minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…In this question you will explore Graph Colouring algorithms. Given a graph G, we say that G is k-colourable if every vertex of G can be assigned one of k colours so that for every pair u, v of adjacent vertices, u and v are assigned different colours. The chromatic number of a graph G, denoted by χ(G), is the smallest integer k for which graph G is k-colorable. To show that χ(G) = k, you must show that the graph is k-colourable and that the graph is not (k − 1)-colourable. Question: It is NP-complete to determine whether an arbitrary graph has chromatic number k, where k ≥ 3. However, determining whether an arbitrary graph has chromatic number 2 is in P. Given a graph G on n vertices, create an algorithm that will return TRUE if χ(G) = 2 and FALSE if χ(G) 6= 2. Clearly explain how your algorithm works, why it guarantees the correct output, and determine the running time of your algorithm.Q2 a Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. 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