Geodesic
Semitotal domination in claw-free graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1585-1605 Cet article a éte moissonné depuis la source Library of Science

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In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by γ_t2(G), is the minimum cardinality of a semitotal dominating set in G. We prove that if G is a connected claw-free graph of order n with minimum degree δ(G)≥ 2 and is not one of fourteen exceptional graphs (ten of which are cycles), then γ_t2(G) ≤37n, and we also characterize the graphs achieving equality, which are an infinite family of graphs. In particular, if we restrict δ(G) ≥ 3 and G K_4, then we can improve the result to γ_t2(G) ≤25n, solving the conjecture for the case of claw-free graphs, proposed by Goddard, Henning and McPillan in [Semitotal domination in graphs, Util. Math. 94 (2014) 67–81].
Keywords: semitotal domination, minimum degree, claw-free graphs 
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Chen, Jie; Liang, Yi-Ping; Xu, Shou-Jun. Semitotal domination in claw-free graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1585-1605. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a18/

[1] J. Chen and S.-J. Xu, A characterization of 3-\gamma-critical graphs which are not bicri-tical, Inform. Process. Lett. 166 (2021) 106062. https://doi.org/10.1016/j.ipl.2020.106062

[2] Y. Dong, E. Shan, L. Kang and S. Li, Domination in intersecting hypergraphs, Discrete Appl. Math. 251 (2018) 155–159. https://doi.org/10.1016/j.dam.2018.05.039

[3] W. Goddard, M.A. Henning and C.A. McPillan, Semitotal domination in graphs, Util. Math. 94 (2014) 67-81.

[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (CRC Press, Boca Raton, 1998). https://doi.org/10.1201/9781482246582

[5] M.A. Henning and A. Yeo, Total Domination in Graphs (Springer, New York, 2013). https://doi.org/10.1007/978-1-4614-6525-6

[6] M.A. Henning, Bounds on domination parameters in graphs: a brief survey, Discuss. Math. Graph Theory 42 (2022) 665–708. https://doi.org/10.7151/dmgt.2454

[7] M.A. Henning and A.J. Marcon, On matching and semitotal domination in graphs, Discrete Math. 324 (2014) 13–18. https://doi.org/10.1016/j.disc.2014.01.021

[8] M.A. Henning, Edge weighting functions on semitotal dominating sets, Graphs Combin. 33 (2017) 403–417. https://doi.org/10.1007/s00373-017-1769-4

[9] M.A. Henning and A. Pandey, Algorithmic aspects of semitotal domination in graphs, Theoret. Comput. Sci. 766 (2019) 46–57. https://doi.org/10.1016/j.tcs.2018.09.019

[10] E. Zhu, Z. Shao and J. Xu, Semitotal domination in claw-free cubic graphs, Graphs Combin. 33 (2017) 1119–1130. https://doi.org/10.1007/s00373-017-1826-z