Perfect edge domination: hard and solvable cases
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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 \(P_5\).
KeywordsClaw-free graphs Complexity dichotomy Cubic graphs NP-completeness Perfect edge domination
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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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. (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