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  1. write an algorithm to find a spanning tree that has the maximum number of leaves.
  2. Write an algorithm to find a spanning tree with minimum number of nodes.

I am yet not able to come up with a solution for the following questions.

For the first part what I thought is to find the vertex with the highest degree and place it in the second last level such that the last level gets the maximum number of leaves.

Prateek Agrawal
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1 Answers1

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  1. Finding a spanning tree of a graph with maximum number of leaves is an NP-Complete problem. There is a reduction from the Dominating Set Problem which is NP-Complete.

  2. Finding a spanning tree of a graph with minimum number of leaves is also an NP-Complete problem. Suppose if the graph has a Hamiltonian path then the graph has a spanning tree with just two leaves. Thus finding a spanning tree of a graph with minimum number of leaves is equivalent to finding whether a graph has a Hamiltonian path or not.

So for both the problems you need to develop approximation algorithms.

Deepu
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