Degree-constrained spanning tree

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In graph theory, a degree-constrained spanning tree is a spanning tree where the maximum vertex degree is limited to a certain constant k. The degree-constrained spanning tree problem is to determine whether a particular graph has such a spanning tree for a particular k. Formally:

Input: n-node undirected graph G(V,E); positive integer k ≤ n.

Question: Does G have a spanning tree in which no node has degree greater than k?

This problem is NP-Complete. This can be shown by a reduction from the Hamiltonian path problem. It remains NP-complete even if k is fixed to a value ≥ 2 (in fact, for k=2, this is the Hamiltonian path problem).

References

Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A2.1: ND1, pg.206.

If you have defined that the degree must be <= k, the k=2 case of degree-confined spanning tree is the Hamiltonian path problem.