Hardness of Approximating Independent Domination in Circle Graphs.
A graph G = (V,E) is called a circle graph if there is a one toone 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 ∈ V eith...
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Language:  English 
Published: 
1999

Online Access:  http://ezproxy.villanova.edu/login?url=https://digital.library.villanova.edu/Item/vudl:175680 
Summary:  A graph G = (V,E) is called a circle graph if there is a one toone 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 ∈ V either u ∈ 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. Keil (Discrete Applied Mathematics, 42 (1993), 51–63) shows that the minimum dominating set problem and the minimum connected dominating set problem are both NPcomplete 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 NPcomplete. Furthermore we show that for any ε, 0 ≤ ε < 1, there does not exist an n^ε approximation algorithm for the minimum independent dominating set problem on nvertex circle graphs, unless P = NP. Several other related domination problems on circle graphs are also shown to be as hard to approximate. 
