DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

Article in press


Authors:

M. Dettlaff

Magda Dettlaff

University of Gdańsk

email: magda.dettlaff@ug.edu.pl

0000-0002-7296-1893

M.A. Henning

Michael A. Henning

University of Johannesburg

email: mahenning@uj.ac.za

0000-0001-8185-067X

J. Topp

Jerzy Topp

The State University of Applied Sciences in Elbląg

email: j.topp@ans-elblag.pl

0000-0002-8069-7850

Title:

Characterization of $\alpha$-excellent $2$-trees

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Source:

Discussiones Mathematicae Graph Theory

Received: 2022-12-31 , Revised: 2023-08-28 , Accepted: 2023-08-29 , Available online: 2023-10-03 , https://doi.org/10.7151/dmgt.2520

Abstract:

A graph is $\alpha$-excellent if every vertex of the graph is contained in some maximum independent set of the graph. In this paper, we present two characterizations of the $\alpha$-excellent $2$-trees.

Keywords:

independence number, excellent graph, $k$-tree

References:

  1. C. Berge, The theory of Graphs and its Applications (Methuen, London, 1962).
  2. C. Berge, Some common properties for regularizable graphs, edge-critical graphs and $B$-graphs, in: Graph Theory and Algorithms, N. Saito, T. Nishiezeki (Ed(s)), Lecture Notes in Comput. Sci. 108, (Springer, Berlin, Heidelberg 1981) 108–123.
    https://doi.org/10.1007/3-540-10704-5_10
  3. C. Berge, Graphs (North-Holland, Amsterdam, 1985).
  4. M. Dettlaff, M.A. Henning and J. Topp, On $\alpha$-excellent graphs, Bull. Malays. Math. Sci. Soc. 46 (2023) 65.
    https://doi.org/10.1007/s40840-022-01456-0
  5. M. Dettlaff, M. Lemańska and J. Topp, Common independence in graphs, Symmetry 13(8) (2021) 1411.
    https://doi.org/10.3390/sym13081411
  6. G.S. Domke, J.H. Hattingh and L.R. Markus, On weakly connected domination in graphs II, Discrete Math. 305 (2005) 112–122.
    https://doi.org/10.1016/j.disc.2005.10.006
  7. G.H. Fricke, T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi and R.C. Laskar, Excellent trees, Bull. Inst. Combin. Appl. 34 (2002) 27–38.
  8. W. Goddard and M.A. Henning, Independent domination in graphs: A survey and recent results, Discrete Math. 313 (2013) 839–854.
    https://doi.org/10.1016/j.disc.2012.11.031
  9. T.W. Haynes, S.T. Hedetniemi and M.A. Henning, Topics in Domination in Graphs, Dev. Math. 64 (Springer, Cham, 2020).
    https://doi.org/10.1007/978-3-030-51117-3
  10. T.W. Haynes, S.T. Hedetniemi and M.A. Henning, Structures of Domination in Graphs, Dev. Math. 66 (Springer, Cham, 2021).
    https://doi.org/10.1007/978-3-030-58892-2
  11. T.W. Haynes, S.T. Hedetniemi and M.A. Henning, Domination in Graphs: Core Concepts, Springer Monogr. Math. (Springer, Cham, 2023).
    https://doi.org/10.1007/978-3-031-09496-5
  12. O. Ore, Theory of Graphs, Amer. Math. Soc. Colloq. Publ. 38 (Providence, RI, 1962).
  13. M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98.
    https://doi.org/10.1016/S0021-9800(70)80011-4
  14. M.D. Plummer, Well-covered graphs: a survey, Quaest. Math. 16 (1993) 253–287.
    https://doi.org/10.1080/16073606.1993.9631737
  15. A.P. Pushpalatha, G. Jothilakshmi, S. Suganthi and V. Swaminathan, $\beta_0$-excellent graphs, Int. J. Contemp. Math. Sci. 6 (2011) 1447–1451.
  16. A.P. Pushpalatha, G. Jothilakshmi, S. Suganthi and V. Swaminathan, Very $\beta_0$-excellent graphs, Taga Journal 14 (2018) 144–148.
  17. D.J. Rose, On simple characterizations of $k$-trees, Discrete Math. 7 (1974) 317–322.
    https://doi.org/10.1016/0012-365X(74)90042-9
  18. J. Topp, Domination, Independence and Irredundance in Graphs, Dissertationes Math. 342 (Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1995).

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