Let e be the unique lightest edge in a graph G. Let T be a spanning tree of G such that e /∈ T . Prove using elementary properties of spanning trees (i.e. not the cut property) that T is not a minimum spanning tree of G.
Let e be the unique lightest edge in a graph G. Let T be a spanning...
Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in T' such that the graph obtained by adding the edge e, to T-e is again a spanning tree in G. Let G be a graph, and let T, T' be spanning trees in G. Show that if e is an edge in T, then there is an edge e in...
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e. Suppose G has n vertices and m edges. Let E’ be a given subset of the edges of E such that the edges of E’ do not form a cycle. (E’ is given as part of input.) Design an O(mlogn) time algorithm for finding a minimum spanning tree of G induced by E’. Prove that your algorithm indeed runs in O(mlogn) time. A minimum...
1) Professor Sabatier conjectures the following converse of Theorem 23.1. Let G=(V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S,V−S) be any cut of G that respects A, and let (u,v) be a safe edge for A crossing (S,V−S). Then, (u,v) is a light edge for the cut. Show that the professor's conjecture is incorrect...
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is increased. The input to your algorithm should be the edge e and its new weight: your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is decreased. The input to your algorithm should be the edge e and its new weight; your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
A bridge is an edge whose removal disconnects the graph. Prove that any bridge must be in some minimum spanning tree. You may use the cut property in the proof, if you want.
Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.
Problem 3's picture are given below. 5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...