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.
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Formal definition
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?
NP-completeness
This problem is NP-complete (Garey & Johnson 1979). 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 the problem is defined as the degree must be ≤ k, the k = 2 case of degree-confined spanning tree is the Hamiltonian path problem.
Degree-constrained minimum spanning tree
On a weighted graph, a Degree-constrained minimum spanning tree (DCMST) is a degree-constrained spanning tree in which the sum of its edges has the minimum possible sum. Finding a DCMST is an NP-Hard problem.
Heuristic algorithms that can solve the problem in polynomial time have been proposed, including Genetic and Ant-Based Algorithms.
Approximation Algorithm
Fürer & Raghavachari (1994) gave an approximation algorithm for the problem which, on any given instance, either shows that the instance has no tree of maximum degree k or it finds and returns a tree of maximum degree k+1.