Annals of Operations Research

, Volume 264, Issue 1–2, pp 287–305 | Cite as

Perfect edge domination: hard and solvable cases

  • Min Chih Lin
  • Vadim Lozin
  • Veronica A. Moyano
  • Jayme L. Szwarcfiter
Original Paper
  • 55 Downloads

Abstract

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset \(E'\) of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of \(E'\). We say that \(E'\) is a perfect edge dominating set of G, if every edge not in \(E'\) is dominated by exactly one edge of \(E'\). The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most \(d \ge 3\) and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a \(P_5\)-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a \(P_5\).

Keywords

Claw-free graphs Complexity dichotomy Cubic graphs NP-completeness Perfect edge domination 

Notes

Acknowledgements

We appreciate the comments of an anonymous reviewer, which significantly helped us improving the presentation and clarity of this work. Min Chih Lin and Veronica A. Moyano were partially supported by UBACyT Grants 20020120100058 and 20020130100800BA, and PICT ANPCyT Grant 2013-2205. Vadim Lozin acknowledges support of the Russian Science Foundation, Grant 17-11-01336. Jayme L. Szwarfiter was partially supported by CNPq and CAPES, research agencies.

References

  1. Bacsó, G., & Tuza, Z. (1990). Dominating cliques in \(P_5\)-free graphs. Periodica Mathemayica Hungarica, 21, 303–308.CrossRefGoogle Scholar
  2. Brandstadt, A., Leitert, A., & Rautenbach, D. (2012). Efficient dominating and edge dominating sets for graphs and hypergraphs. In Proceedings of the 23rd international symposium on algorithms and computation (ISAAC 2012), Lecture Notes in Computer Science (Vol. 7676, pp. 558–277). .Google Scholar
  3. Brandstadt, A., & Mosca, R. (2011). Dominating induced matching for \(P_7\)-free graphs in linear time. In Proceedings of the 22nd international symposium on algorithms and computation (ISAAC 2011), Lecture Notes in Computer Science (pp. 100–109).Google Scholar
  4. Cardoso, D. M., Cerdeira, J. O., Delorme, C., & Silva, P. C. (2008). Efficient edge domination in regular graphs. Discrete Applied Mathematics, 156, 3060–3065.CrossRefGoogle Scholar
  5. Camby, E., & Schaudt, O. (2014). A new characterization of \(P_k\)-free graphs, Graph-theoretic concepts in computer science—40th international workshop (WG 2014), France, Revised Selected Papers (pp. 129–138).Google Scholar
  6. Cardoso, D. M., Koperlainen, N., & Lozin, V. V. (2011). On the complexity of the induced matching problem in hereditary classes of graphs. Discrete Applied Mathematics, 159, 521–531.CrossRefGoogle Scholar
  7. Georges, J. P., Halsey, M. D., Sanaulla, A. M., & Whittlesey, M. A. (1990). Edge domination and graph structure. Congressus Numerantium, 76, 127–144.Google Scholar
  8. Grinstead, D. L., Slater, P. J., Sherwani, N. A., & Holnes, N. D. (1993). Efficient edge domination problems in graphs. Information Processing Letters, 48, 221–228.CrossRefGoogle Scholar
  9. Hertz, A., Lozin, V., Ries, B., Zamaraev, V., & de Werra, D. (2015). Dominating induced matchings in graphs containing no long claw. Journal of Graph Theory (accepted).Google Scholar
  10. Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2013a). An \(O^*(1.1939^n)\) time algorithm for minimum weighted dominating induced matching. In Proceedings of the 24th international symposium on algorithms and computation (ISAAC 2013), Hong Kong, Lecture Notes in Computer Science (Vol. 8283, pp. 558–567).Google Scholar
  11. Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2013b). Exact algorithms for dominating induced matching. Corr arXiv:1301.7602
  12. Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2014). Fast algorithms for some dominating induced matching problems. Information Processing Letters, 114, 524–528.CrossRefGoogle Scholar
  13. Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2015a). Efficient and perfect domination on circular-arc graphs. In Proceedings of the VIII Latin-American graphs, algorithms and optimization symposium (LAGOS’ 2015), Beberibe, Brazil, Electronic Notes in Discrete Mathematics (to appear).Google Scholar
  14. Lin, M. C., Moyano, V., Rautenbach, D., & Szwarcfiter, J. L. (2015b). The maximum number of dominating induced matchings. Journal of Graph Theory, 78, 258–268.CrossRefGoogle Scholar
  15. Lu, C. L., Ko, M.-T., & Tang, C. Y. (2002). Perfect edge domination and efficient edge domination in graphs. Discrete Applied Mathematics, 119, 227–250.CrossRefGoogle Scholar
  16. Lu, C. L., & Tang, C. Y. (1998). Solving the weighted efficient edge domination problem in bipartite permutation graphs. Discrete Applied Mathematics, 87, 203–211.CrossRefGoogle Scholar
  17. Xiao, M., & Nagamochi, H. (2015). Exact algorithms for dominating induced matching based on graph partition. Discrete Applied Mathematics, 190–191, 147–162.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Consejo Nacional de Investigaciones Científicas y TécnicasBuenos AiresArgentina
  2. 2.Instituto de Cálculo and Departamento de ComputaciónUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.University of WarwickCoventryUK
  4. 4.Instituto de Cálculo and Departamento de MatemáticaUniversidad de Buenos AiresBuenos AiresArgentina
  5. 5.I. Mat., COPPE and NCEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  6. 6.Instituto Nacional de Metrologia, Qualidade e TecnologiaRio de JaneiroBrazil

Personalised recommendations