Question: Let G = (V, E) be an undirected weighted graph with distinct weights we for every edge e E. Let T(V, E) denote the edge

Let G = (V, E) be an undirected weighted graph with

Let G = (V, E) be an undirected weighted graph with distinct weights we for every edge e E. Let T(V, E) denote the edge set of the unique MST of (V, E). Let W(T) = Eect We be the weight of the edge set T. We say an edge e is necessary for the Minimum Spanning Tree T(G, E) if when we delete it, the weight of the MST increases. So we have that W (T(V, E \ {e})) > W(T(V, E))). Show that an edge e is necessary if and only if for every cycle C in G that contains it, e is not the maximum weight edge in this cycle (so there exists e' EC such that we > w.). Let G = (V, E) be an undirected weighted graph with distinct weights we for every edge e E. Let T(V, E) denote the edge set of the unique MST of (V, E). Let W(T) = Eect We be the weight of the edge set T. We say an edge e is necessary for the Minimum Spanning Tree T(G, E) if when we delete it, the weight of the MST increases. So we have that W (T(V, E \ {e})) > W(T(V, E))). Show that an edge e is necessary if and only if for every cycle C in G that contains it, e is not the maximum weight edge in this cycle (so there exists e' EC such that we > w.)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Databases Questions!

Let G = (V, E) be an undirected, fully-connected graph with real weights that are all distinct (i.e., no two edges have the same weight). Let T' be the minimum spanning tree of G and let T'' be a...

Algorithm analysis / let G = (V, E) be an undirected weighted graph, and let T be the shortest-path spanning tree rooted at a vertex v. Suppose now that all the edge weights in G are increased by a...

PLEASE do not just copy/paste the answer for this question which is already provided on here by someone else, that answer is insufficient. Let G = (V, E) be an undirected connected graph and let c :...

) Let G = (V, E) be an undirected connected graph and let c : E R + be a function specifying the costs of the edges, i.e., every edge has a positive cost. Assume that no two edges have the same cost....

Let G = (V. E) be an undirected graph with all edge weights equal to I. Let d(i, j) denote the length of the shortest path between i and j. Now, suppose we wish to answer queries from the user of the...

Let G = (V, E) be an undirected, connected graph with weight function w : E R, and suppose that |E| |V| and all edge weights are distinct. A second-best minimum spanning tree is defined as follows....

Question 3. (1 marks) Let G = (V, E) be an undirected connected graph with n nodes and m edges. In this question you are asked to orient the graph, i.e. pick one direction for each edge: an edge (u....

1. Let G = (V, E, w) be a directed graph with weighted edges w : E R; edge weights could be positive, negative, or zero. We're going to design another algorithm for computing all-pairs shortest...

Let G = (V, E) be an undirected connected graph and let c : E R + be a function specifying the costs of the edges, i.e., every edge has a positive cost. Assume that no two edges have the same cost....

1 Dalol Company, which began operations on January 2, Year 4, appropriately uses the installment method of accounting. The following information is available for the year ended December 31, Year 4....

Figure 4-38(b) shows a second alternative data model for the sheet-music-tracking problem. This alternative shows two variations on the Work entity. In the second variation, an attribute named Work...

indicate the way each item below should be listed 1 . commerical paper 2 . noncommited line of credit

Seved Help 14 Wisconsin Snowmobile Corp. is considering a switch to level production Cost efficiencies would occur under level production, and aftertax costs would decline by $31,500, but inventory...

Assume that the banking system has total reserves of $100 billion. Assume also that required reserves are 10 percent of checking deposits and that banks hold no excess reserves and households hold no...

As shown in Figure 3, the overall labor-force participation rate of men declined between 1970 and 2000. At the same time, the labor-force participation rate of women increased sharply. This overall...

The Bureau of Labor Statistics announced that in February 2008, of all adult Americans, 145,993,000 were employed, 7,381,000 were unemployed, and 79,436,000 were not in the labor force. Use this...