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