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Perfect edge domination: hard and solvable cases

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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 \(P_5\).

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References

  • Bacsó, G., & Tuza, Z. (1990). Dominating cliques in \(P_5\)-free graphs. Periodica Mathemayica Hungarica, 21, 303–308.

    Article  Google Scholar 

  • 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). .

  • 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).

  • Cardoso, D. M., Cerdeira, J. O., Delorme, C., & Silva, P. C. (2008). Efficient edge domination in regular graphs. Discrete Applied Mathematics, 156, 3060–3065.

    Article  Google Scholar 

  • 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).

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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).

  • 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).

  • Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2013b). Exact algorithms for dominating induced matching. Corr arXiv:1301.7602

  • Lin, M. C., Mizrahi, M., & Szwarcfiter, J. L. (2014). Fast algorithms for some dominating induced matching problems. Information Processing Letters, 114, 524–528.

    Article  Google Scholar 

  • 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).

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lu, C. L., & Tang, C. Y. (1998). Solving the weighted efficient edge domination problem in bipartite permutation graphs. Discrete Applied Mathematics, 87, 203–211.

    Article  Google Scholar 

  • Xiao, M., & Nagamochi, H. (2015). Exact algorithms for dominating induced matching based on graph partition. Discrete Applied Mathematics, 190–191, 147–162.

    Article  Google Scholar 

Download references

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.

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Lin, M.C., Lozin, V., Moyano, V.A. et al. Perfect edge domination: hard and solvable cases. Ann Oper Res 264, 287–305 (2018). https://doi.org/10.1007/s10479-017-2664-3

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