## Abstract

In a graph G= (V, E), a non-empty set A of k distinct vertices, is called a k-attack on G. The vertices in the set A is considered to be under attack. A set D⊆ V can defend or counter the attack A on G if there exists a one to one function f: A⟼ D, such that either f(u) = u or there is an edge between u and it’s image f(u), in G. A set D is called a k-defensive dominating set, if it defends against any k-attack on G. Given a graph G= (V, E), the minimum k-defensive domination problem requires us to compute a minimum cardinality k-defensive dominating set of G. When k is not fixed, it is co-NP-hard to decide if D⊆ V is a k-defensive dominating set. However, when k is fixed, the decision version of the problem is NP-complete for general graphs. On the positive side, the problem can be solved in linear time when restricted to paths, cycles, co-chain graphs and threshold graphs for any k. In this paper, we mainly focus on the problem when k> 0 is fixed. We prove that the decision version of the problem remains NP-complete for bipartite graphs, this answers a question asked by Ekim et al. (Discrete Math. 343 (2) (2020)). We give lower and upper bound on the approximation ratio for the problem. Further, we show that the minimum k-defensive domination problem is APX-complete for bounded degree graphs. On the positive side, we show that the problem is efficiently solvable for complete bipartite graphs for any k> 0.

Original language | English |
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Title of host publication | Combinatorial Optimization and Applications - 15th International Conference, COCOA 2021, Proceedings |

Editors | Ding-Zhu Du, Donglei Du, Chenchen Wu, Dachuan Xu |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 247-261 |

Number of pages | 15 |

ISBN (Print) | 9783030926809 |

DOIs | |

Publication status | Published - 2021 |

Event | 15th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2021 - Tianjin, China Duration: 17 Dec 2021 → 19 Dec 2021 |

### 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 | 13135 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 15th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2021 |
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Country/Territory | China |

City | Tianjin |

Period | 17/12/21 → 19/12/21 |

## Keywords

- APX-completeness
- Approximation algorithms
- Defensive domination
- Domination
- Graph algorithms
- NP-completeness

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science