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Semitotal Domination in Claw-Free Cubic Graphs

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Abstract

In this paper, we continue the study of semitotal domination in graphs in [Discrete Math. 324, 13–18 (2014)]. A set \({S}\) of vertices in \({G}\) 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, \({{\gamma_{t2}}(G)}\), is the minimum cardinality of a semitotal dominating set of \({G}\). This domination parameter is squeezed between arguably the two most important domination parameters; namely, the domination number, \({\gamma (G)}\), and the total domination number, \({{\gamma_{t}}(G)}\). We observe that \({\gamma (G) \leq {\gamma_{t2}}(G) \leq {\gamma_{t}}(G)}\). A claw-free graph is a graph that does not contain \({K_{1, \, 3}}\) as an induced subgraph. We prove that if \({G}\) is a connected, claw-free, cubic graph of order \({n \geq 10}\), then \({{\gamma_{t2}}(G) \leq 4n/11}\).

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References

  1. Chudnovsky M., Seymour P.: Claw-free graphs. V. Global structure. J. Combin. Theory Ser. B 98(6), 1373–1410 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cockayne E.J., Favaron O., Mynhardt C.M.: Total domination in claw-free cubic graphs. J. Combin. Math. Combin. Comput. 43, 219–225 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Faudree R., Flandrin E., Ryjáček Z.: Claw-free graphs — a survey. Discrete Math. 164(1-3), 87–147 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Favaron O., Henning M.A.: Paired-domination in claw-free cubic graphs. Graphs Combin. 20(4), 447–456 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Favaron O., Henning M.A.: Total domination in claw-free graphs with minimum degree 2. Discrete Math. 308(15), 3213–3219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Favaron O., Henning M.A.: Bounds on total domination in claw-free cubic graphs. Discrete Math. 308(16), 3491–3507 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goddard W., Henning M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839–854 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Goddard W., Henning M.A., McPillan C.A.: Semitotal domination in graphs. Util. Math. 94, 67–81 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Haynes T.W., Hedetniemi S.T., Slater P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)

    MATH  Google Scholar 

  10. Haynes T.W., Hedetniemi S.T., Slater P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, Inc., New York (1998)

    MATH  Google Scholar 

  11. Haynes T.W., Henning M.A., Howard J.: Locating and total dominating sets in trees. Discrete Appl. Math. 154(8), 1293–1300 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Henning M.A.: A survey of selected recent results on total domination in graphs. Discrete Math. 309(1), 32–63 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henning M.A., Kang L., Shan E., Yeo A.: On matching and total domination in graphs. Discrete Math. 308(11), 2313–2318 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henning M.A., Löwenstein C.: Locating-total domination in claw-free cubic graphs. Discrete Math. 312(21), 3107–3116 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henning M.A., Marcon A.J.: On matching and semitotal domination in graphs. Discrete Math. 324, 13–18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Henning M.A., Marcon A.J.: Vertices contained in all or in no minimum semitotal dominating set of a tree. Discuss. Math. Graph Theory 36(1), 71–93 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henning M.A., Yeo A.: Hypergraphs with large transversal number and with edge sizes at least three. J. Graph Theory 59(4), 326–348 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Henning M.A., Yeo A.: Total Domination in Graphs. Springer, New York (2013)

    Book  MATH  Google Scholar 

  19. Lichiardopol N.: On a conjecture on total domination in claw-free cubic graphs: proof and new upper bound. Australas. J. Combin. 51, 7–28 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Southey J., Henning M.A.: On a conjecture on total domination in claw-free cubic graphs. Discrete Math. 310(21), 2984–2999 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

    Michael A. Henning & Alister J. Marcon

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  1. Michael A. Henning
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  2. Alister J. Marcon
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Correspondence to Michael A. Henning.

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Research supported in part by the South African National Research Foundation and the University of Johannesburg.

Funded by the South African National Research Foundation.

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Henning, M.A., Marcon, A.J. Semitotal Domination in Claw-Free Cubic Graphs. Ann. Comb. 20, 799–813 (2016). https://doi.org/10.1007/s00026-016-0331-z

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