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Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that for all u, v ∈ V , the path between u and v in tree T is unique
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- Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that the graph G need not have a unique tree TLet G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that a) for all u, v € V, the path between u and u in tree T is unique b) the graph G need not have a unique tree TLet G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.
- 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').G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proof
- Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setBe G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
- Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.Which 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.1. Prove the following known result on minimum spanning trees. Theorem. Let G = (V, E) be a graph and let w : E → R+ be a weight function. Let A be a subset of edges that is contained in some minimum spanning tree, and let (S, S) be a cut so that (S, S)N A = 0. Prove that if e is a light edge in (S, S), then e is safe for A. %3D 2. Let S = fi and a profit p; associated with it. Design a dynamic programing algorithm for computing a set of compatible activities of maximum profit. You should start with defining some terms and then write a recurrence relation for solving the problem. What is the time complexity of your algorithm? {1,2...n} be a set activities, where activity i E S has a start time si, a finish time