Abstract
Every graph is the edge intersection graph of subtrees of a tree. The tree-degree of a graph is the minimum maximal degree of the underlying tree for which there exists a subtree intersection model. Computing the tree-degree is NP-complete even for planar graphs, but polynomial time algorithms exist for outer-planar graphs, diamond-free graphs and chordal graphs. The number of minimal separators of graphs with bounded tree-degree is polynomial. This implies that the treewidth of graphs with bounded tree-degree can be computed efficiently, even without the model given in advance.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnborg, S., D. G. Corneil, and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Discrete methods 8, (1987), pp. 277–284.
Berry, A., J.-P. Bordat, O. Cogis, Generating all the minimal separators of a graph, Internat. J. Found. Comput. Sci. 11 (2000), pp. 397–403.
Booth, K. S., G. S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Comput. Syst. Sci. 13 (1976), pp. 335–379.
Bodlaender, H. L., D. M. Thilikos, Tree width of graphs with small chordality, Discrete Applied Mathematics 79, (1997), pp. 45–61.
Bouchitté, V., I. Todinca, Minimal triangulations for graphs with “few” minimal separators, Proceedings ESA’98, LNCS 1461, Springer, 1998, pp. 344–355.
Brandstädt, A., V. B. Le, and J. P. Spinrad, Graph-classes—A Survey, SIAM monographs on discrete mathematics and application, Philadelphia, (1999).
Breu, H., D. G. Kirkpatrick, Unit disk graph recognition is NP-hard, Comput. Geom. 9, (1998), pp. 3–24.
Erdős, P., A. Goodman, L. Pósa, The representation of graphs by set intersection, Cand. J. Math. 18, (1966), pp. 106–112.
Gavril, F., The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory 16, (1974), pp. 47–56.
Golumbic, M. C., Algorithmic graph theory and perfect graphs, Academic press, NewY ork, (1980).
Golumbic, M. C., R. E. Jamison, Edge and vertex intersections of paths in trees, Discrete Math. 55, (1995), pp. 151–159.
Golumbic, M. C., R. E. Jamison, The edge-intersection graphs of paths in a tree, J. Comb. Theory B 38, (1985), pp. 8–22
Holyer, I., The NP-completeness of some edge-partition problems, SIAM J. Comput. 4, 1981, pp. 713–717.
Hoover, D. N., Complexity of graph covering problems for graphs of lowdegree, JCMCC 11, 1992, pp. 187–208.
Kou, L. T., L. J. Stockmeyer and C. K. Wong, Covering edges by cliques with regard to keyword conflicts and intersection graphs, Comm. ACM 21, 1978, pp. 135–139.
Kratochvíl, J., String graphs II. Recognising string graphs in NP-hard, J. Comb. Theory B, 52, (1991), pp. 67–78.
Lovász, L., On coverings of graphs, in: P. Erdős and G. Katona eds., Proceedings of the Colloquium held at Tihany, Hungary, 1966, Academic Press, New York, 1968, pp. 231–236.
Ma, S., W. D. Wallis and J. Wu, Clique covering of chordal graphs, Utilitas Mathematica 36, (1989), pp. 151–152.
Orlin, J., Contentment in graph theory, Proc. of the Nederlandse Academie van Wetenschappen, Amsterdam, Series A, 80, 1977, pp. 406–424.
Pullman, N. J., Clique covering of graphs IV. Algorithms, SIAM J. Comput. 13, (1984), pp. 57–75.
Tarjan, R. E., Decomposition by clique separators, Discrete Mathematics 55, (1985), pp. 221–223.
Uehara, R., NP-complete problems on a 3-connected cubic planar graph and their application, Technical report TWCU-M-0004, Tokyo 1996. http://www.komazawa-u.ac.jp/~uehara/ps/triangle.ps.gz
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chang, MS., Müller, H. (2001). On the Tree-Degree of Graphs. In: Brandstädt, A., Le, V.B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2001. Lecture Notes in Computer Science, vol 2204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45477-2_6
Download citation
DOI: https://doi.org/10.1007/3-540-45477-2_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42707-0
Online ISBN: 978-3-540-45477-9
eBook Packages: Springer Book Archive