Abstract
Given a graph \(G=(V, E)\), a \(P_2\)-packing \(\mathcal {P}\) is a collection of vertex disjoint copies of \(P_2\)s in \(G\) where a \(P_2\) is a simple path with three vertices and two edges. The Maximum \(P_2\)-Packing problem is to find a \(P_2\)-packing \(\mathcal {P}\) in the input graph \(G\) of maximum cardinality. This problem is NP-hard for cubic graphs. In this paper, we give a branch-and-reduce algorithm for the Maximum \(P_2\)-Packing problem in cubic graphs. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time \(O^{*}(1.4366^n)\) which is faster than previous known exact algorithms where \(n\) is the number of vertices in the input graph.
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Acknowledgments
This research is partially supported by the Ministry of Science and Technology of Taiwan under Grants NSC 101–2221–E–241–019–MY3 and NSC 102–2221–E–241–007–MY3. Ling-Ju Hung (corresponding author) is supported by the Ministry of Science and Technology of Taiwan under Grant NSC 103–2811–E–241–001.
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Parts of this paper have been presented in Proceedings of ICS 2014: Workshop on Algorithms and Computation Theory.
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Chang, MS., Chen, LH. & Hung, LJ. An \(O^{*}(1.4366^n)\)-time exact algorithm for maximum \(P_2\)-packing in cubic graphs. J Comb Optim 32, 594–607 (2016). https://doi.org/10.1007/s10878-015-9884-8
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DOI: https://doi.org/10.1007/s10878-015-9884-8