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ISAAC 1999: Algorithms and Computation pp 70-82

# Constant-Factor Approximation Algorithms for Domination Problems on Circle Graphs

• Mirela Damian-Iordache
• Sriram V. Pemmaraju
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)

## Abstract

A graph G = (V,E) is called a circle graph if there is a one- to-one correspondence between vertices in V and a set C of chords in a circle such that two vertices in V are adjacent if and only if the corre- sponding chords in C intersect. A subset V′ of V is a dominating set of G if for all uV either uV′ or u has a neighbor in V′. In addition, if G[V′] is connected, then V′ is called a connected dominating set; if G[V′] has no isolated vertices, then V′ is called a total dominating set. Keil (Discrete Applied Mathematics, 42 (1993), 51–63) shows that the minimum dominating set problem (MDS), the minimum connected dominating set problem (MCDS) and the minimum total domination problem (MTDS) are all NP-complete even for circle graphs. He mentions designing approximation algorithms for these problems as being open. This paper presents O(1)-approximation algorithms for all three problems — MDS, MCDS, and MTDS on circle graphs. For any circle graph with n vertices and m edges, these algorithms take O(n 2 + nm) time and O(n 2) space. These results, along with the result on the hardness of approximating minimum independent dominating set on circle graphs (Damian-Iordache and Pemmaraju, in this proceedings) advance our understanding of domination problems on circle graphs significantly.

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## References

1. 1.
E. S. Elmallah, L. K. Stewart, and J. Culberson. Polynomial algorithms on k-polygon graphs. In Proceedings of the 21st Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton (Florida), 1990.Google Scholar
2. 2.
D. S. Johnson. The NP-completeness column: an ongoing guide. Journal of Algorithms, 6:434–451, 1985.
3. 3.
J. Mark Keil. The complexity of domination problems in circle graphs. Discrete Applied Mathematics, 42:51–63, 1991.
4. 4.
M.R. Garey and D.S. Johnson. Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company, New York, 1979.

## Copyright information

© Springer-Verlag Berlin Heidelberg 1999

## Authors and Affiliations

• Mirela Damian-Iordache
• 1
• Sriram V. Pemmaraju
• 2
1. 1.Department of Computer ScienceUniversity of IowaIowa CityUSA
2. 2.Department of MathematicsIndian Institute of TechnologyBombay, PowaiIndia