Degree-constrained spanning tree

On the left, a spanning tree can be constructed where the vertex with the highest degree is 2 (thus, a max degree 2 tree).
On the right, the central vertex must have degree at least 5 in any tree spanning this graph, so a 2 degree constrained tree cannot be constructed here.

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.

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) give an iterative polynomial time algorithm which, given a graph $G$, returns a spanning tree with maximum degree no larger than $\Delta ^{*}+1$, where $\Delta ^{*}$ is the minimum possible maximum degree over all spanning trees. Thus, if $k=\Delta ^{*}$, such an algorithm will either return a spanning tree of maximum degree $k$ or $k+1$.

References

• Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, ISBN 978-0-7167-1045-5. 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, CiteSeerX 10.1.1.136.1089, doi:10.1006/jagm.1994.1042.