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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 uniqueLet 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 TBe 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 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').(a) Let Ge (v, E) be an undinected let each edge e e E have weight weightle), and suppese all edge weights Then, the edge of minimum weight in connected graph, IVI >3 and are diffesent. Let T be a minimum spanning tree in. G. G must belang to T. an undirected graph G=(V,E), IVI ン3, has connect are minimum spanning tree, then all the edge weights unique よ5ferent. FOLLOWING STATEMET PROOF THE Oe SHOW A COUNTEREXAMPLElet 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 connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)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.
- Part 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…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.(V, E) be an undirected graph. Assume that the vertices 2. Let G represent documents. The edges are weighted so that w(i, j) is the dissimilarity between documents i and j. It is desired to partition the vertices into k > 3 disjoint clusters such that k ΣΣ υ(u, υ). i=1 (u,v)EE u,vEC; is minimized. The set C; is the set of documents in cluster i. Show that the e-approximate version of this problem is P-hard for all e, e > 0. Note that k is a fixed integer provided with each problem instance and may be different for different instances.