Abstract
The semitotal domination number of a graph G without isolated vertices is the minimum cardinality of a set S of vertices of G such that every vertex in \(V(G){\setminus } S\) is adjacent to at least one vertex in S, and every vertex in S is within distance 2 of another vertex of S. In Henning and Marcon (Ann Comb 20(4):1–15, 2016), it was shown that every connected claw-free cubic graph G of order n has semitotal domination number at most \(\frac{4n}{11}\) when \(n\ge 10\), and it was conjectured that the bound can be improved from \(\frac{4n}{11}\) to \(\frac{n}{3}\) if \(G\notin \{K_4,N_2\}\). In this paper, we prove this conjecture.
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Acknowledgements
Many thanks to the anonymous referees for their careful comments that improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grants 61672051, 61309015, and the Fundamental Research Funds for the Central Universities of China under Grant DUT16RC(3)065.
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Zhu, E., Shao, Z. & Xu, J. Semitotal Domination in Claw-Free Cubic Graphs. Graphs and Combinatorics 33, 1119–1130 (2017). https://doi.org/10.1007/s00373-017-1826-z
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DOI: https://doi.org/10.1007/s00373-017-1826-z