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A Characterization of 3-(γ c , 2)-Critical Claw-Free Graphs Which are not 3-γ c -Critical

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Abstract

Let γ c (G) denote the minimum cardinality of a connected dominating set for G. A graph G is k-γ c -critical if γ c (G) = k, but γ c (G + xy) < k for \({xy \in E(\overline {G})}\) . Further, for integer r ≥ 2, G is said to be k-(γ c , r)-critical if γ c (G) = k, but γ c (G + xy) < k for each pair of non-adjacent vertices x and y that are at distance at most r apart. k-γ c -critical graphs are k-(γ c , r)-critical but the converse need not be true. In this paper, we give a characterization of 3-(γ c , 2)-critical claw-free graphs which are not 3-γ c -critical. In fact, we show that there are exactly four classes of such graphs.

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References

  1. Ananchuen, N.: On local edge connected domination critical graphs. Util. Math. (2010) (in press)

  2. Ananchuen N.: On domination critical graphs with cutvertices having connected domination number 3. Int. Math. Forum 2, 3041–3052 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Ananchuen N., Plummer M.D.: Some results related to the toughness of 3-domination critical graphs. Discrete Math. 272, 5–15 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ananchuen N., Plummer M.D.: Matching properties in domination critical graphs. Discrete Math. 277, 1–13 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ananchuen N., Plummer M.D.: Some results related to the toughness of 3-domination critical graphs II. Util. Math. 70, 11–32 (2006)

    MATH  MathSciNet  Google Scholar 

  6. Ananchuen N., Plummer M.D.: 3-factor-criticality in domination critical graphs. Discrete Math. 307, 3006–3015 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ananchuen N., Ananchuen W., Plummer M.D.L.: Matching properties in connected domination critical graphs. Discrete Math. 308, 1260–1267 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bondy J.A., Murty U.S.R.: Graph theory with applications. The Macmillan Press, London (1976)

    Google Scholar 

  9. Chen Y., Edwin Cheng T.C., Ng C.T.: Hamilton-connectivity of 3-domination critical graphs with α = δ + 1 ≥ 5. Discrete Math. 308, 1296–1307 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen X-G., Sun L., Ma D-X.: Connected domination critical graphs. Appl. Math. Lett. 17, 503–507 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen Y., Tian F.: A new proof of Wojcicka’s conjecture. Discrete Appl. Math. 127, 545–554 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Flandrin E., Tian F., Wei B., Zhang L.: Some properties of 3-domination-critical graphs. Discrete Math. 205, 65–76 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Henning M.A., Oellermann O.R., Swart H.C.: Local edge domination critical graphs. Discrete Math. 161, 175–184 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sumner D.P., Blitch P.: Domination critical graphs. J. Comb. Theory Ser. B 34, 65–76 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sumner D.P., Wojcicka E.: Graphs critical with respect to the domination number. In: Haynes, T.W., Hedetniemi, S.T., Slater, P. (eds) Domination in graphs: advanced topics, pp. 439–469. Marcel Dekker Inc, New York (1998)

    Google Scholar 

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Authors and Affiliations

  1. School of Liberal Arts, Sukhothai Thammathirat Open University, Pakkred, Nonthaburi, 11120, Thailand

    Watcharaphong Ananchuen

  2. Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom, 73000, Thailand

    Nawarat Ananchuen

  3. Western Australian Centre of Excellence in Industrial Optimisation, Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia

    Louis Caccetta

Authors
  1. Watcharaphong Ananchuen
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  2. Nawarat Ananchuen
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  3. Louis Caccetta
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Corresponding author

Correspondence to Nawarat Ananchuen.

Additional information

W. Ananchuen and N. Ananchuen supported by the Thailand Research Fund grant # BRG50800017.

L. Caccetta supported by the Western Australian Centre of Excellence in Industrial Optimisation (WACEIO).

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Ananchuen, W., Ananchuen, N. & Caccetta, L. A Characterization of 3-(γ c , 2)-Critical Claw-Free Graphs Which are not 3-γ c -Critical. Graphs and Combinatorics 26, 315–328 (2010). https://doi.org/10.1007/s00373-010-0920-2

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  • DOI: https://doi.org/10.1007/s00373-010-0920-2

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