  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.
