### Abstract

The paper studies a generalization of the Independent Set (IS) problem. A distance-d independent set for a positive integer d ≥ 2 in an unweighted graph G = (V, E) is a set S ⊆ V of vertices such that for any pair of vertices u, v ∈ S, the distance between u and v is at least d in G. Given an unweighted graph G and a positive integer k, the Distance- d Independent Set (D d IS) problem is to decide whether G contains a distance-d independent set S such that |S| ≥ k. D2IS is identical to the original IS and thus D2IS is in for bipartite graphs and chordal graphs. In this paper, we show that for every fixed integer d ≥ 3, D d IS is -complete even for planar bipartite graphs of maximum degree three, and also -complete even for chordal bipartite graphs. Furthermore, we show that if the input graph is restricted to chordal graphs, then D d IS can be solved in polynomial time for any even d ≥ 2, whereas D d IS is -complete for any odd d ≥ 3.

Original language | English |
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Title of host publication | Combinatorial Optimization and Applications - 6th International Conference, COCOA 2012, Proceedings |

Pages | 234-244 |

Number of pages | 11 |

DOIs | |

Publication status | Published - Aug 20 2012 |

Externally published | Yes |

Event | 6th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2012 - Banff, AB, Canada Duration: Aug 5 2012 → Aug 9 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7402 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2012 |
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Country | Canada |

City | Banff, AB |

Period | 8/5/12 → 8/9/12 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Combinatorial Optimization and Applications - 6th International Conference, COCOA 2012, Proceedings*(pp. 234-244). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7402 LNCS). https://doi.org/10.1007/978-3-642-31770-5_21