Theory of Computing Systems

, Volume 54, Issue 1, pp 45–72

# Parameterized Domination in Circle Graphs

• Nicolas Bousquet
• Daniel Gonçalves
• George B. Mertzios
• Christophe Paul
• Ignasi Sau
• Stéphan Thomassé
Article

## Abstract

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:
• Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution.

• Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs.

• If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $$2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$$, where n=|V(G)|.

## Keywords

Circle graphs Domination problems Parameterized complexity Parameterized algorithms Dynamic programming Constrained domination

## Notes

### Acknowledgements

We would like to thank Sylvain Guillemot for stimulating discussions that motivated some of the research carried out in this paper, and to the anonymous referees for helpful suggestions that improved the presentation of the manuscript.

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## Authors and Affiliations

• Nicolas Bousquet
• 1
• Daniel Gonçalves
• 1
• George B. Mertzios
• 2
• Christophe Paul
• 1
• Ignasi Sau
• 1
• Stéphan Thomassé
• 3
1. 1.AlGCo project-teamCNRS, LIRMMMontpellierFrance
2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
3. 3.Laboratoire LIPU. Lyon, CNRS, ENS Lyon, INRIA, UCBLLyonFrance