The low-degree spanning tree problem is as follows. Given a graph G and an integer k,does G contain a spanning tree such that all vertices in the tree have degree at most k(obviously, only tree edges count towards the degree)? For example, in the following graph,there is no spanning tree such that all vertices have a degree at most three.(a) Prove that the low-degree spanning tree problem is NP-hard with a reduction fromHamiltonian path.(b) Now consider the high-degree spanning tree problem, which is as follows. Givena graph G and an integer k, does G contain a spanning tree whose highest degreevertex is at least k? In the previous example, there exists a spanning tree with ahighest degree of 7. Give an efficient algorithm to solve the high-degree spanningtree problem, and an analysis of its time complexity.

(a) To prove that the low-degree spanning tree problem is NP-hard, we can reduce the Hamiltonian…

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