Abstract
The neighborhood packing number of a graph is the maximum number of pairwise vertex disjoint closed neighborhoods in the graph. This number is a lower bound on the domination number of the graph. We show that the domination number of a graph of girth at least 7 is bounded from above by a (quadratic) function of its closed neighborhood packing number, and further that no such bound exists for graphs of girth at most 6. We then show that as girth of the graph increases, the upper bound on the domination number drops as a function of girth.
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference no. 819416.
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Madathil, J., Misra, P., Saurabh, S. (2019). An Erdős–Pósa Theorem on Neighborhoods and Domination Number. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_36
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DOI: https://doi.org/10.1007/978-3-030-26176-4_36
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