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 from Hamiltonian path. (b) Now consider the high-degree spanning tree problem, which is as follows. Given a graph G and an integer k, does G contain a spanning tree whose highest degree vertex is at least k? In the previous example, there exists a spanning tree with a highest degree of 7. Give an efficient algorithm to solve the high-degree spanning tree problem, and an analysis of its time complexity.

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 from Hamiltonian path. (b) Now consider the high-degree spanning tree problem, which is as follows. Given a graph G and an integer k, does G contain a spanning tree whose highest degree vertex is at least k? In the previous example, there exists a spanning tree with a highest degree of 7. Give an efficient algorithm to solve the high-degree spanning tree problem, and an analysis of its time complexity.