Degree-constrained spanning tree

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.

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.

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.

References

• Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, ISBN 0-7167-1045-5. A2.1: ND1, p. 206.{{citation}}: CS1 maint: postscript (link)

• Fürer, Martin; Raghavachari, Balaji (1994), "Approximating the minimum-degree Steiner tree to within one of optimal", Journal of Algorithms, 17 (3): 409–423, doi:10.1006/jagm.1994.1042.