Abstract
The contribution of this work is to show that the recently-proposed primeval and homogeneous decompositions of graphs can be used to solve efficiently various types of weighted domination and Steiner tree problems. Furthermore, we point out that these results imply linear-time algorithms for large classes of graphs which, in some local sense, contain only a small number of induced P 4s.
This is a preview of subscription content, log in via an institution to check access.
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
Babel, L., Olariu, S.: On the structure of graphs with few P4s. Discrete Applied Mathematics 84, 1–13 (1998)
Baumann, S.: A linear algorithm for the homogeneous decomposition of graphs. Report No. M-9615. Zentrum Mathematik, Technische Universität München (1996)
Corneil, D.G., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3, 163–174 (1981)
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9, 27–39 (1984)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14, 926–934 (1985)
Cournier, A., Habib, M.: A new linear time algorithm for modular decomposition. Trees in Algebra and Programming. LNCS, vol. 787, pp. 68–84. Springer, Heidleberg (1994)
Dahlhaus, E., Gustedt, J., McConnell, R.: Efficient and practical modular decomposition. In: Eight Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, Louisiana , pp. 26–35 (1997)
Haynes, T., Hedetniemi, S., Slater, P. (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)
Hedetniemi, S., Laskar, R.: Topics on Domination. Annals of Discrete Mathematics, vol. 48. North-Holland, Amsterdam (1991)
Jamison, B., Olariu, S.: P 4-reducible graphs, a class of uniquely tree representable graphs. Studies in Applied Mathematics 81, 79–87 (1989)
Jamison, B., Olariu, S.: On a unique tree representation for P4-extendible graphs. Discrete Applied Mathematics 34, 151–164 (1991)
Jamison, B., Olariu, S.: A unique tree representation for P 4-sparse graphs. Discrete Applied Mathematics 35, 115–129 (1992)
Jamison, B., Olariu, S.: p-components and the homogeneous decomposition of graphs. SIAM J. Discrete Mathematics 8, 448–463 (1995)
Kratsch, D., Stewart, L.: Domination on cocomparability graphs. SIAM J. Discrete Mathematics 6, 400–417 (1993)
Laskar, R., Pfaff, J., Hedetniemi, S.M., Hedetniemi, S.T.: On the algorithmic complexity of total domination. SIAM J. Algebraic Discrete Methods 5, 420–425 (1984)
Lin, R., Olariu, S.: A fast parallel algorithm to recognize P4-sparse graphs. Discrete Applied Mathematics 81, 191–215 (1998)
McConnell, R., Spinrad, J.: Linear-time modular decomposition and efficient transitive orientation of comparability graphs. In: Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Arlington, VA, pp. 536–545 (1994)
Möhring, R.H.: Algorithmic aspects of comparability graphs and interval graphs. In: Rival, I. (ed.) Graphs and Orders. Holland, Dordrecht (1985)
White, K., Farber, M., Pulleyblank, W.: Steiner trees, connected domination and strongly chordal graphs. Networks 15, 109–124 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Babel, L., Olariu, S. (1998). Domination and Steiner Tree Problems on Graphs with Few P 4s. In: Hromkovič, J., Sýkora, O. (eds) Graph-Theoretic Concepts in Computer Science. WG 1998. Lecture Notes in Computer Science, vol 1517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10692760_27
Download citation
DOI: https://doi.org/10.1007/10692760_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65195-6
Online ISBN: 978-3-540-49494-2
eBook Packages: Springer Book Archive