# Bounded degree spanning trees

## Abstract

Given a connected graph *G*, let a *Δ*_{T}-spanning tree of *G* be a spanning tree of *G* of maximum degree bounded by *Δ*_{T}. It is well known that for each *Δ*_{T} ≥ 2 the problem of deciding whether a connected graph has a *Δ*_{T}-spanning tree is *NP*-complete. In this paper we investigate this problem when additionally connectivity and maximum degree of the graph are given. A complete characterization of this problem for 2- and 3-connected graphs, for planar graphs, and for *Δ*_{T} = 2 is provided.

Our first result is that given a biconnected graph of maximum degree 2*Δ*_{T} - 2, we can find its *Δ*_{T}-spanning tree in time *O*(*m + n*^{3/2}). For graphs of higher connectivity we design a polynomial-time algorithm that finds a *Δ*_{T}-spanning tree in any *k*-connected graph of maximum degree *k*(*Δ*_{T} − 2) + 2. On the other hand, we prove that deciding whether a *k*-connected graph of maximum degree *k*(*Δ*_{T} - 2) + 3 has a *Δ*_{T}-spanning tree is *NP*-complete, provided *k* ≤ 3. For arbitrary *k* ≥ 3 we show that verifying whether a *k*-connected graph of maximum degree *k*(*Δ*_{T} - 1) has a *Δ*_{T}-spanning tree is *NP*-complete. In particular, we prove that the Hamiltonian path (cycle) problem is *NP*-complete for *k*-connected *k*-regular graphs, if *k* > 2. This extends the well known result for *k* = 3 and fully characterizes the case *Δ*_{T} = 2.

For planar graphs it is NP-complete to decide whether a k-connected planar graph of maximum degree *Δ*_{G} has a *Δ*_{T}-spanning tree for *k* = 1 and *Δ*_{g} > *Δ*_{T} ≥ 2, for *k* = 2 and *Δ*_{G} > 2(*Δ*_{T} -1) ≥ 2, and for *k* = 3 and *Δ*_{G} > *Δ*_{T} = 2. On the other hand, we show how to find in polynomial (linear or almost linear) time a *Δ*_{T}-spanning tree for all other parameters of *k*, *Δ*_{G}, and *Δ*_{T}.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D. Barnette. Trees in polyhedral graphs.
*Canadian J. Mathematics*, 18:731–736, 1966.Google Scholar - 2.D. Bauer, S. L. Hakimi, and E. F. Schmeichel. Recognizing tough graphs is
*NP*-hard.*Discrete Applied Mathematics*, 28:191–195, 1990.Google Scholar - 3.B. Bollobás.
*Extremal Graph Theory*. Academic Press, London, 1978.Google Scholar - 4.P. M. Camerini, G. Galgiati, and R. Maffioli. Complexity of spanning tree problems, I.
*European Journal of Operation Research*, 5:346–352, 1980.Google Scholar - 5.Y. Caro, I. Krasikov, and Y. Roditty. On the largest tree of a given maximum degree in a connected graph.
*Journal of Graph Theory*, 15:7–13, 1991.Google Scholar - 6.N. Chiba and T. Nishizeki. The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs.
*Journal of Algorithms*, 10:187–211, 1989.Google Scholar - 7.M. Fairer and B. Raghavachari. Approximating the minimum-degree Steiner tree to within one of optimal.
*Journal of Algorithms*, 17:409–423,1994. Also in ACM-SIAM SODA 1992.Google Scholar - 8.M. R. Garey and D. S. Johnson.
*Computers and Intractability: A Guide to the Theory of NP-completeness*. Freeman, New York, 1979.Google Scholar - 9.M. R. Garey, D. S. Johnson, and R. E. Tarjan. The planar Hamiltonian circuit problem is
*NP*-complete.*SIAM Journal on Computing*, 5(4):704–714, 1976.Google Scholar - 10.M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems.
*SIAM Journal on Computing*, 24(2):296–317, 1995.Google Scholar - 11.H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse
*k*-connected spanning subgraph of a*k*-connected graph.*Algorithmica*, 7:583–596, 1992.Google Scholar - 12.B. Jackson and T. D. Parsons. On
*r*-regular*r*-connected non-Hamiltonian graphs.*Bulletin of Australian Mathematics Society*, 24:205–220, 1981.Google Scholar - 13.D. S. Johnson. The
*NP*-completeness column: An ongoing guide.*Journal of Algorithms*, 6:434–451, 1985.Google Scholar - 14.S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. In
*Proceedings of the 27 ACM STOC*, pp. 1–10, 1995.Google Scholar - 15.S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight.
*SIAM Journal on Computing*, 25(2):355–368, 1996.Google Scholar - 16.S. Khuller and U. Vishkin. Biconnectivity approximations and graph carvings.
*Journal of the ACM*, 41(2):214–235, 1994.Google Scholar - 17.T. Lukovski and W.-B. Strothmann. Decremental biconnectivity on planar graphs. Manuscript, 1997.Google Scholar
- 18.G. H. J. Meredith. Regular
*n*-valent*n*-connected nonHamiltonian non-n-edge-colorable graphs.*Journal of Combinatorial Theory Series B*, 14:55–60, 1973.Google Scholar - 19.V. Neumann-Lara and E. Rivera-Campo. Spanning trees with bounded degrees.
*Combinatorica*, 11(1):55–61, 1991.Google Scholar - 20.C. H. Papadimitriou and M. Yannakakis. The complexity of restricted spanning tree problems.
*Journal of the ACM*, 29(2):285–309, 1982.Google Scholar - 21.M. Rauch. Improved data structures for fully dynamic biconnectivity. Full version. A preliminary version appeared in
*Proceedings of the 26th ACM STOC*, 1994.Google Scholar - 22.M. Rauch Henzinger and V King. Fully dynamic biconnectivity and transitive closure. In
*Proceedings of the 36th IEEE FOCS*, pp. 664–673, 1995.Google Scholar - 23.D.D. Sleator and R.E. Tarjan. A data structure for dynamic trees.
*Journal of Computer and System Sciences*26:362–391, 1983.Google Scholar - 24.W.-B. Strothmann. Constructing 3-trees in 3-connected planar graphs in linear time. Manuscript, 1996.Google Scholar
- 25.S. Win. On a connection between the existence of k-trees and the toughness of a graph.
*Graphs and Combinatorics*, 5:201–205, 1989.Google Scholar