Consider the following graph: Which of the following does not describe the graph? O It is a connected graph. O The graph has nine (9) edges. O The graph's vertex set can be partitioned into two subsets V1={1,2,3} and V2={4,5,6} where every edge connects a vertex from V1 and a vertex from V2. ○ It is the complete graph with six vertices, i.e., K6. O It is the complete bipartite graph K3,3-
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- Given a vertex set W = {1, 2, 3, 4}, solve for the following questions: (i) Knowing that edges are the same if and only if they have the same endpoint, how many different edges are possible? (ii) Suppose for this question that graphs are different if they have different sets of edges (they do not depend on if the graphs are isomorphic or not). How many simple graphs are there with the vertex set W? (iii) Draw the eleven nonisomorphic simple graphs that have four vertices.Consider the following graph. If we run breadth first search on this graph starting at any vertex, which one of the following is a possible order for visiting the nodes? NQMPOR QMNPOR MNOPQR QMNPRO Assume the following adjacency list for a graph. After you draw the graph, what is the path to reach D from A. Vertex Edges A --------> B, C B --------> C C --------> B, E D --------> B E --------> B, D Both (B) and (C) A->B->C->E->D A->B->C->D A->C->E->Dlet a graph have vertices h,i,j,k,l,m,n,o and edge set {{h,i},{h,j},{h,k},{h,n},{h,o},{j,k},{j,l},{j,m},{k,l},{m,o},. on paper, draw the graph. then answer the questions below. a. what is the degree of vertex k? b. what is the degree of vertex h? c. how many connected components does the graph have?
- 5. (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…Answer True or False to the following claims: a. If G is graph on at most 5 vertices and every vertex has degree 2, then G is a cycle. b. Let G be a forest. If we add an edge to G, then G is no longer a forest. c. Let G be a graph, and let u, v, and w be vertices in G. Suppose that a shortest path from u to v in G is of length 3, andsuppose that a shortest path from v to w in G is of length 4. Then a shortest path from u to w in G is of length 7.How many undirected graphs (not necessarily connected) can be constructed out of a given set V= {V 1, V 2,…V n} of n vertices ? Group of answer choices 2^(n(n-1)/2) 2^n n(n-l)/2 n! Which of the following is an advantage of adjacency list representation over adjacency matrix representation of a graph? Adding a vertex in adjacency list representation is easier than adjacency matrix representation. In adjacency list representation, space is saved for sparse graphs. DFS and BSF can be done in O(V + E) time for adjacency list representation. These operations take O(V^2) time in adjacency matrix representation. Here is V and E are number of vertices and edges respectively. All of the above Given the starting vertex A, what is the visit order of the graph shown in Fig. 1 under the DFS traversal algorithm. ABDCFE ACBDFE ABCDFE ADBCEF Assume you have the adjacency matrix representing a graph. 1 represents a connection while -1 represents a lack of one:[-1, 1, - 1][-1, -1, 1][1, -1,…
- In the game of Nim, an arbitrary number of chips are divided into an arbitrary number of piles. Each player can remove as many chips as desired from any single pile. The last player to remove a chip wins. Consider a limited version of this game, in which three piles contain 3, 5, and 8 chips, respectively. You can represent this game as a directed graph. Each vertex in this graph is a possible configuration of the piles (chips in each pile). The initial configuration, for example, is (3, 5, 8). Each edge in the graph represents a legal move in the game. Write Java statements that will construct this directed graph. Discuss how a computer program might use this graph to play Nim. java programThe clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: (Image taken from the Wikipedia 'Clique Problem' page.) You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a 4-clique.…The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: (Image taken from the Wikipedia 'Clique Problem' page.) You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a 4-clique.…
- Let G be a graph with 7 vertices, where eachvertex is labelled by a number from 1 to 7. Twovertices are selected at random. Let us call themu and v. Now, directed edges are drawn from u toall other vertices except v and directed edges aredrawn from all vertices to v except from u. Let xbe the total possible topological sortings of G.hint: You may remember the terms factorials orfibonacci?Write down the value of x:Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v] 5. Write the full pseudocode for the following problem using BFS/DFS.Input: An undirected graph G that’s not necessarily connected.Output: Is G bipartite? That is, can the vertices of G be partitioned into two sets X, and Y such that thereare no edges within X and Y ?Given the following set of vertices V and set of edges E. V = {p, q, r, s, t, u} E = {(p,q), (q,r), (r, s), (t,p), (q,t), (r, t), (t,u)} Which of the following is correct? Select one: A.There are 7 vertices B.There are 7 edges. C.The degree at vertex p is 7 D.The total degree of all vertices is 7