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Graphs and Combinatorics

, Volume 35, Issue 4, pp 881–912 | Cite as

Independent Domination in Bipartite Cubic Graphs

  • Christoph BrauseEmail author
  • Michael A. Henning
Original Paper
  • 34 Downloads

Abstract

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set. In this paper, we study the following conjecture posed by Goddard and Henning (Discrete Math. 313:839–854, 2013): If \(G\not \cong K_{3,3}\) is a connected, cubic, bipartite graph on n vertices, then \(i(G) \le \frac{4}{11}n\). Henning et al. (Discrete Appl. Math. 162:399–403, 2014) prove the conjecture when the girth is at least 6. In this paper we strengthen this result by proving the conjecture when the graph has no subgraph isomorphic to \(K_{2,3}\).

Keywords

Domination Independent Domination Cubic Graphs 

Mathematics Subject Classification

05C69 

Notes

Acknowledgements

The authors express their sincere thanks to the referees for their meticulous and thorough reading of the paper.

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(2), 205–208 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dorbec, P., Henning, M.A., Montassier, M., Southey, J.: Independent domination in cubic graphs. J. Graph Theory 80(4), 329–349 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313, 839–854 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Goddard, W., Lyle, J.: Independent dominating sets in triangle-free graphs. J. Combin. Optim. 23(1), 9–20 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goddard, W., Henning, M.A., Lyle, J., Southey, J.: On the independent domination number of regular graphs. Ann. Combin. 16, 719–732 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)zbMATHGoogle Scholar
  7. 7.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, Inc., New York (1998)zbMATHGoogle Scholar
  8. 8.
    Henning, M.A., Löwenstein, C., Rautenbach, D.: Independent domination in subcubic bipartite graphs of girth at least six. Discrete Appl. Math. 162, 399–403 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kostochka, A.V.: The independent domination number of a cubic \(3\)-connected graph can be much larger than its domination number. Graphs Combin. 9(3), 235–237 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lam, P.C.B., Shiu, W.C., Sun, L.: On independent domination number of regular graphs. Discrete Math. 202, 135–144 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lyle, J.: A note on independent sets in graphs with large minimum degree and small cliques. Electron. J. Combin. 21(2), 1–17 (2014). (Paper #P2.38)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Suil, O., West, D.B.: Cubic graphs with large ratio of independent domination mumber to domination number. Graphs Combin. 32(2), 773–776 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Southey, J.: Domination results: Vertex partitions and edge weight functions. Ph.D Thesis, University of Johannesburg, May (2012)Google Scholar
  14. 14.
    Southey, J., Henning, M.A.: Domination versus independent domination in cubic graphs. Discrete Math. 313, 1212–1220 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and AlgebraTU Bergakademie FreibergFreibergGermany
  2. 2.Department of Mathematics and Applied MathematicsUniversity of JohannesburgAuckland ParkSouth Africa

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