Summary: 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 \(u\in V\) either \(u\in V'\) 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. J. Keil [Discrete Appl. Math. 42, No. 1, 51-63 (1993; Zbl 0786.05079)] 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 \(\varepsilon\), \(0\leq \varepsilon< 1\), there does not exist an \(n^\varepsilon\)-approximation algorithm for the minimum independent dominating set problem on \(n\)-vertex circle graphs, unless \(\text{P}= \text{NP}\). Several other related domination problems on circle graphs are also shown to be as hard to approximate.
For the entire collection see [Zbl 0933.00045].