# Hardness of Approximating Independent Domination in Circle Graphs

## 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 corresponding chords in *C* intersect. A subset *V′* of *V* is a *dominating set* of *G* if for all *u* ∈ *V* either *u* ∈ *V′* or *u* has a neighbor in *V′*. In addition, if no two vertices in *V′* are adjacent, then *V′* is called an *independent dominating set*; if *G*[*V′*] is connected, then *V′* is called a *connected dominating set*. Keil (*Discrete Applied Mathematics*, 42 (1993), 51–63) shows that the minimum dominating set problem and the minimum connected dominating set problem are both NP-complete even for circle graphs. He leaves open the complexity of the minimum independent dominating set problem. In this paper we show that the minimum independent dominating set problem on circle graphs is NP-complete. Furthermore we show that for any ε, 0 ≤ ε < 1, there does not exist an *n* ^{ ε }-approximation algorithm for the minimum independent dominating set problem on *n*-vertex circle graphs, unless P = NP. Several other related domination problems on circle graphs are also shown to be as hard to approximate.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Sanjeev Arora and Carsten Lund. Hardness of approximation. In Dorit S. Hochbaum, editor,
*Approximation Algorithms for NP-hard problems*. PWS Publishing Company, 1997.Google Scholar - 2.T. A. Beyer, A. Proskurowski, S.T. Hedetniemi, and S. Mitchell. Independent domination in trees.
*Congressus Numerantium*, 19:321–328, 1977.MathSciNetGoogle Scholar - 3.K. S. Booth. Dominating sets in chordal graphs. Technical Report CS-80-34, Department of Computer Science, University ofWaterloo,Waterloo, Ontario, 1980.Google Scholar
- 4.K. S. Booth and J. H. Johnson. Dominating sets in chordal graphs.
*SIAM Journal on Computing*, 13:335–379, 1976.zbMATHGoogle Scholar - 5.C. J. Colbourn and L. K. Stewart. Permutation graphs: connected domination and Steiner trees.
*Discrete Mathematics*, 86:179–189, 1990.zbMATHCrossRefMathSciNetGoogle Scholar - 6.D. G. Corneil and Y. Perl. Clustering and domination in perfect graphs.
*Discrete Applied Mathematics*, 9:27–39, 1984.zbMATHCrossRefMathSciNetGoogle Scholar - 7.Mirela Damian-Iordache and Sriram V. Pemmaraju. Domination in circle graphs with applications to polygon decomposition. Technical Report 99-02, University of Iowa, 1999.Google Scholar
- 8.A. K. Dewdney. Fast turing reductions between problems in NP, chapter 4: Reductions between NP-complete problems. Technical Report 71, Department of Computer Science, University of Western Ontario, London, Ontario, 1983.Google Scholar
- 9.M. Farber. Independent domination in chordal graphs.
*Operations Research Letters*, 1:134–138, 1982.zbMATHCrossRefMathSciNetGoogle Scholar - 10.M. Farber and J. M. Keil. Domination in permutation graphs.
*Journal of Algorithms*, 6:309–321, 1985.zbMATHCrossRefMathSciNetGoogle Scholar - 11.T. W. Haynes, S. T. Hedetniemi, and P. J. Slater.
*Fundamentals of Domination in Graphs*. Number 208 in Pure and Applied Mathematics: A series of monomgraphs and textbooks. Marcel Dekker Inc., New York, 1998.Google Scholar - 12.Robert W. Irving. On approximating the minimum independent dominating set.
*Information Processing Letters*, 37:197–200, 1991.zbMATHCrossRefMathSciNetGoogle Scholar - 13.D. S. Johnson. The NP-completeness column: an ongoing guide.
*Journal of Algorithms*, 6:434–451, 1985.zbMATHCrossRefMathSciNetGoogle Scholar - 14.J. Mark Keil. The complexity of domination problems in circle graphs.
*Discrete Applied Mathematics*, 42:51–63, 1991.CrossRefGoogle Scholar - 15.D. Kratsch and L. Stewart. Domination on cocomparability graphs.
*SIAM Journal on Discrete Mathematics*, 6(3):400–417, 1993.zbMATHCrossRefMathSciNetGoogle Scholar - 16.R. Laskar and J. Pfaff. Domination and irredundance in split graphs. Technical Report 430, Clemson University, 1983.Google Scholar
- 17.Y. D. Liang. Steiner set and connected domination in trapezoid graphs.
*Information Processing Letters*, 56:101–108, 1995.CrossRefMathSciNetGoogle Scholar - 18.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.zbMATHGoogle Scholar - 19.J. Pfaff, R. Laskar, and S. T. Hedetniemi. NP-completeness of total and connected domination, and irredundance for bipartite graphs. Technical Report 428, Clemson University, 1983.Google Scholar