. Let G = (V,E) be a dominating pair graph and {u, v} any domi nating pair with distance greater than 4. Then the shortest paths in I are MCDS of G.
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1. Let G = (V,E) be a dominating pair graph and {u, v} any domi nating pair with distance greater than 4. Then the shortest paths in I are MCDS of G.
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- 1. Let G = (V, E) be a dominating pair graph and {u, v} any domi- nating pair with distance greater than 4. Then the shortest paths in I are MCDS of G.Consider a connected graph G with at least 4 edges that has all distinct edge weights. Which of the following properties must be true of a Minimum Spanning Tree (MST) of G? I. The MST must contain the shortest edge of G. II. The MST must contain the second-shortest edge of G. III. The MST can never contain the longest edge of G. None I Only I and II Only I and III Only O1, II, and III4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of G, and for each vertex v, we know its pre and post numbers. Show the following: (a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there is a directed path from u to v in G. (b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there is no directed path from u to v in G.
- Consider a connected graph G with at least 4 edges that has all distinct edge weights. Which of the following properties must be true of a Minimum Spanning Tree (MST) of G? I. The MST must contain the shortest edge of G. II. The MST must contain the second-shortest edge of G. III. The MST can never contain the longest edge of G. O None OI Only OI and II Only I and III Only OI, II, and IIILet w be the minimum weight among all edge weights in an un-directed connected graph. Let e be a specific edge of weight w. Which of the following statements is/are TRUE/FALSE? Please discuss each of these statements on why that statement is True/False. a. Every minimum spanning tree has an edge of weight wb. If e is not in a minimum spanning tree T, then in the cycle formed by adding e to T, all edges have the same weightc. There is a minimum spanning tree containing e.d. e is present in every minimum spanning treeWhich is true/false. Let G be a connected undirected weighted graph. Consider the following two statements. S₁: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree.
- Let G = (V, E) be an undirected graph with vertices V and edges E. Let w(e) denote the weight of e E E. Let T C E be a spanning tree of G. Select all of the following that imply that T is not a minimum spanning tree (MST) for G. Incorrect choices will be penalized. There exists e' (u, v) g T, u, v E V such that w(e') w(e'). O There exists e' g T such that w(e') w(e) for all e E E. O There exists e' (u, v) É T, u, v E V such that w(e') < w(e) for all e on the shortest path from u to v in T. O There exists e E T, e' ¢ T with w(e) < w(e').Consider a connected undirected graph G=(V,E) in which every edge e∈E has a distinct and nonnegative cost. Let T be an MST and P a shortest path from some vertex s to some other vertex t. Now suppose the cost of every edge e of G is increased by 1 and becomes ce+1. Call this new graph G′. Which of the following is true about G′ ? a) T must be an MST and P must be a shortest s - t path. b) T must be an MST but P may not be a shortest s - t path. c) T may not be an MST but P must be a shortest s - t path. d) T may not be an MST and P may not be a shortest s−t path. Pls use Kruskal's algorithm to reason about the MST.Question 1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G).Prove that for any subset of vertices S, S is a vertex cover in G if and only if V\S is a clique in C.Note: this is an if and only if proof, i.e. you need to show both directions for full credit. Question 2 Part 4.1 implies the following result (which you may use without proof): G has a vertex cover of size at most k if and only if the complement of G has a clique of size at least n−k.Use this fact to give a reduction from VertexCover to Clique. Your solution should have the following two steps:i) First, show the reduction: specify how the inputs to VertexCover, G and k, can be transformed to a valid input pair, H and l, for Clique. Make sure to explain why this takes polynomial time.ii) Second, show that the answer to Clique(H,l) can be converted to the answer of VertexCover(G,k). One possibility is to explain how a YES answer to…
- . Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.12 C F 4 1 A 11 10 7 8 3 B 9. D 2](1) T F Given a directed graph G and a vertex v in the graph, breath first search (BFS) can be used to detect if v is part of a cycle in the graph. (2) T F Let P be a shortest path from some vertex s to some other vertex t in a directed graph. If the weight of each edge in the graph is decreased by one, then P will still be a shortest path from s to t. (3) T F edge Kruskal's algorithm is always correct even in graphs with negative weights. (4) T F For any flow network, there is only one unique way to assign flow value to the edges so as to achieve the maximum flow for the network. NP problems are those problems that cannot be solved in polynomial (5) T F time.