Approximation Algorithm and Hardness Results for Defensive Domination in Graphs

Michael A. Henning, Arti Pandey, Vikash Tripathi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 languageEnglish
Title of host publicationCombinatorial Optimization and Applications - 15th International Conference, COCOA 2021, Proceedings
EditorsDing-Zhu Du, Donglei Du, Chenchen Wu, Dachuan Xu
PublisherSpringer Science and Business Media Deutschland GmbH
Pages247-261
Number of pages15
ISBN (Print)9783030926809
DOIs
Publication statusPublished - 2021
Event15th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2021 - Tianjin, China
Duration: 17 Dec 202119 Dec 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume13135 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference15th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2021
Country/TerritoryChina
CityTianjin
Period17/12/2119/12/21

Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

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