ISAAC 1999: Algorithms and Computation pp 56-69

Hardness of Approximating Independent Domination in Circle Graphs

• Mirela Damian-Iordache
• Sriram V. Pemmaraju
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)

Abstract

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 uV either uV′ 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 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 ε, 0 ≤ ε < 1, there does not exist an n ε -approximation algorithm for the minimum independent dominating set problem on n-vertex circle graphs, unless P = NP. Several other related domination problems on circle graphs are also shown to be as hard to approximate.

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