# Parameterized Domination in Circle Graphs

• Nicolas Bousquet
• Daniel Gonçalves
• George B. Mertzios
• Christophe Paul
• Ignasi Sau
• Stéphan Thomassé
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7551)

## Abstract

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 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 has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

## Keywords

circle graphs domination problems parameterized complexity parameterized algorithms dynamic programming constrained domination

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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 UniversityU.K.
3. 3.Laboratoire LIPU. Lyon, CNRS, ENS Lyon, INRIA, UCBLLyonFrance