# Constant-Factor Approximation Algorithms for Domination Problems on Circle Graphs

## 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 corre- sponding 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 *G*[*V′*] is connected, then *V′* is called a *connected dominating set*; if *G*[*V′*] has no isolated vertices, then *V′* is called a *total dominating set*. Keil (*Discrete Applied Mathematics*, 42 (1993), 51–63) shows that the minimum dominating set problem (MDS), the minimum connected dominating set problem (MCDS) and the minimum total domination problem (MTDS) are all NP-complete even for circle graphs. He mentions designing approximation algorithms for these problems as being open. This paper presents *O*(1)-approximation algorithms for all three problems — MDS, MCDS, and MTDS on circle graphs. For any circle graph with *n* vertices and *m* edges, these algorithms take *O*(*n* ^{2} + *nm*) time and *O*(*n* ^{2}) space. These results, along with the result on the hardness of approximating minimum independent dominating set on circle graphs (Damian-Iordache and Pemmaraju, *in this proceedings*) advance our understanding of domination problems on circle graphs significantly.

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