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# Parameterized domination in circle graphs.

Bousquet, N. and Gonçalves, D. and Mertzios, G.B. and Paul, C. and Sau, I. and Thomassé, S. (2014) 'Parameterized domination in circle graphs.', Theory of computing systems., 54 (1). pp. 45-72.

## 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 2O(t⋅loglogtlogt)⋅nO(1), where n=|V(G)|.

Item Type: Article Circle graphs, Domination problems, Parameterized complexity, Parameterized algorithms, Dynamic programming, Constrained domination. (AM) Accepted Manuscript Download PDF (231Kb) Peer-reviewed http://dx.doi.org/10.1007/s00224-013-9478-8 The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-013-9478-8. No date available 16 September 2014 January 2014 No date available