Which of the following option is false regarding spanning tree?
(A) In an adjacency matrix for an undirected graph with 5 nodes. If all diagonal element is 0 and all non-diagonal element is 1 then there will be multiple distinct MSTs of cost 4.
(B) If we construct a graph by doubling the weight of every edge in the original graph, then we need Ω(E) time to compute the cost of the MST of the new graph.
(C) In an undirected graph G with distinct edge weight if e_max is the edge with maximum weight and e_min is the edge with minimum weight. If e_max is in a minimum spanning tree, then its removal must disconnect G.
(D) In a weighted undirected graph with distinct edges, if every edge weight is increased by same weight then minimum spanning tree is not changed.


Answer: (B)

Explanation: Consider sorting the edge by weight. Doubling the edge weight will not change the sorting order. But this means that Prim\’s and Kruskal will do the same thing so, the MST is unchanged. Therefore the weight of new tree is simply double from old tree and can be computed in constant time if the original weight and MST are known.

In an undirected graph G with distinct edge weight if e_max is the edge with maximum weight and e_min is the edge with minimum weight then G has unique minimum spanning tree, If e_max is in a minimum spanning tree, then its removal must disconnect G, every minimum spanning tree of G must contain e_min but its not necessary that no minimum spanning tree contains e_max.

So, only option (B) is false.

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