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Cubic Graphs with Large Ratio of Independent Domination Number to Domination Number

Abstract

A dominating set in a graph \(G\) is a set \(S\) of vertices such that every vertex outside \(S\) has a neighbor in \(S\); the domination number \(\gamma (G)\) is the minimum size of such a set. The independent domination number, written \(i(G)\), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured \(i(G)/\gamma (G) \le 6/5\) for every cubic (3-regular) graph \(G\) with sufficiently many vertices. We provide an infinite family of counterexamples, giving for each positive integer \(k\) a 2-connected cubic graph \(H_k\) with \(14k\) vertices such that \(i(H_k)=5k\) and \(\gamma (H_k)=4k\).

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References

  1. 1.

    Barefoot, C., Harary, F., Jones, K.F.: What is the difference between the domination and independent domination numbers of a cubic graph? Graphs Combin. 7, 205–208 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    Cockayne, E.J., Hedetniemi, S.T.: Independence and domination in 3-connected cubic graphs. J. Combin. Math. Combin. Comput. 10, 173–182 (1991)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Favaron, O.: Two relations between the parameters of independence and irredundance. Discr. Math. 70, 17–20 (1988)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Gimbel, J., Vestergaard, P.: Inequalities for total matchings of graphs. Ars Combin. 39, 109–119 (1995)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Goddard, W., Henning, M.A., Lyle, J., Southey, J.: On the independent domination number of regular graphs. Annals of Comb. 16, 719–732 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. 6.

    Henning, M.A., Southey, J.: Domination versus independent domination in cubic graphs. Discr. Math. 313, 1212–1220 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. 7.

    Kostochka, A.V.: The independent domination number of a cubic 3-connected graph can be much larger than its domination number. Graphs Combin. 9, 235–237 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. 8.

    Lam, P.C.B., Shiu, W.C., Sun, L.: On independent domination number of regular graphs. Discr. Math. 202, 135–144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Douglas B. West.

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Research supported by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Administration of Foreign Experts Affairs, China.

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O, S., West, D.B. Cubic Graphs with Large Ratio of Independent Domination Number to Domination Number. Graphs and Combinatorics 32, 773–776 (2016). https://doi.org/10.1007/s00373-015-1580-z

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Keywords

  • Independent domination number
  • Domination number
  • Cubic graph
  • 3-regular