On the domination of triangulated discs

Noor A'lawiah Abd Aziz; Nader Jafari Rad; Hailiza Kamarulhaili

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 555-560
  • ISSN: 0862-7959

Abstract

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Let be a -connected triangulated disc of order with the boundary cycle of the outer face of . Tokunaga (2013) conjectured that has a dominating set of cardinality at most . This conjecture is proved in Tokunaga (2020) for being a tree. In this paper we prove the above conjecture for being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.

How to cite

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Abd Aziz, Noor A'lawiah, Jafari Rad, Nader, and Kamarulhaili, Hailiza. "On the domination of triangulated discs." Mathematica Bohemica 148.4 (2023): 555-560. <http://eudml.org/doc/299557>.

@article{AbdAziz2023,
abstract = {Let $G$ be a $3$-connected triangulated disc of order $n$ with the boundary cycle $C$ of the outer face of $G$. Tokunaga (2013) conjectured that $G$ has a dominating set of cardinality at most $\frac\{1\}\{4\}(n+2)$. This conjecture is proved in Tokunaga (2020) for $G-C$ being a tree. In this paper we prove the above conjecture for $G-C$ being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.},
author = {Abd Aziz, Noor A'lawiah, Jafari Rad, Nader, Kamarulhaili, Hailiza},
journal = {Mathematica Bohemica},
keywords = {domination; double domination; total domination; double total domination; planar graph; triangulated disc},
language = {eng},
number = {4},
pages = {555-560},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the domination of triangulated discs},
url = {http://eudml.org/doc/299557},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Abd Aziz, Noor A'lawiah
AU - Jafari Rad, Nader
AU - Kamarulhaili, Hailiza
TI - On the domination of triangulated discs
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 555
EP - 560
AB - Let $G$ be a $3$-connected triangulated disc of order $n$ with the boundary cycle $C$ of the outer face of $G$. Tokunaga (2013) conjectured that $G$ has a dominating set of cardinality at most $\frac{1}{4}(n+2)$. This conjecture is proved in Tokunaga (2020) for $G-C$ being a tree. In this paper we prove the above conjecture for $G-C$ being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.
LA - eng
KW - domination; double domination; total domination; double total domination; planar graph; triangulated disc
UR - http://eudml.org/doc/299557
ER -

References

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  1. Bermudo, S., Sanchéz, J. L., Sigarreta, J. M., 10.1007/s10998-017-0191-2, Period. Math. Hung. 75 (2017), 255-267. (2017) Zbl1413.05279MR3718519DOI10.1007/s10998-017-0191-2
  2. Blidia, M., Chellali, M., Haynes, T. W., 10.1016/j.disc.2006.03.061, Discrete Math. 306 (2006), 1840-1845. (2006) Zbl1100.05068MR2251565DOI10.1016/j.disc.2006.03.061
  3. Cabrera-Martínez, A., 10.1016/j.dam.2022.03.022, Discrete Appl. Math. 315 (2022), 97-103. (2022) Zbl07516299MR4407663DOI10.1016/j.dam.2022.03.022
  4. Cabrera-Martínez, A., Rodríguez-Velázquez, J. A., 10.1016/j.dam.2021.05.011, Discrete Appl. Math. 300 (2021), 107-111. (2021) Zbl1465.05124MR4264160DOI10.1016/j.dam.2021.05.011
  5. Campos, C. N., Wakabayashi, Y., 10.1016/j.dam.2012.08.023, Discrete Appl. Math. 161 (2013), 330-335. (2013) Zbl1254.05136MR2998434DOI10.1016/j.dam.2012.08.023
  6. Fernau, H., Rodríguez-Velázquez, J. A., Sigarreta, J. M., Global powerful -alliances and total -domination in graphs, Util. Math. 98 (2015), 127-147. (2015) Zbl1343.05115MR3410888
  7. Harant, J., Henning, M. A., 10.7151/dmgt.1256, Discuss. Math., Graph Theory 25 (2005), 29-34. (2005) Zbl1073.05049MR2152046DOI10.7151/dmgt.1256
  8. Harary, F., Haynes, T. W., Double domination in graphs, Ars Comb. 55 (2000), 201-213. (2000) Zbl0993.05104MR1755232
  9. Haynes, T. W., Hedetniemi, S. T., Slater, P. J., 10.1201/9781482246582, Pure and Applied Mathematics, Marcel Dekker 208. Marcel Dekker, New York (1998). (1998) Zbl0890.05002MR1605684DOI10.1201/9781482246582
  10. Henning, M. A., Rad, N. Jafari, 10.1007/s00373-020-02249-7, Graphs Comb. 37 (2021), 325-336. (2021) Zbl1459.05239MR4197383DOI10.1007/s00373-020-02249-7
  11. Henning, M. A., Kazemi, A. P., 10.1016/j.dam.2010.01.009, Discrete Appl. Math. 158 (2010), 1006-1011. (2010) Zbl1210.05097MR2607047DOI10.1016/j.dam.2010.01.009
  12. Henning, M. A., Yeo, A., 10.1137/090777001, SIAM J. Discrete Math. 24 (2010), 1336-1355. (2010) Zbl1221.05254MR2735927DOI10.1137/090777001
  13. King, E. L. C., Pelsmajer, M. J., 10.1016/j.disc.2010.03.022, Discrete Math. 310 (2010), 2221-2230. (2010) Zbl1203.05120MR2659172DOI10.1016/j.disc.2010.03.022
  14. Li, Z., Zhu, E., Shao, Z., Xu, J., 10.1016/j.dam.2015.06.024, Discrete Appl. Math. 198 (2016), 164-169. (2016) Zbl1327.05263MR3426889DOI10.1016/j.dam.2015.06.024
  15. Matheson, L. R., Tarjan, R. E., 10.1006/eujc.1996.0048, Eur. J. Comb. 17 (1996), 565-568. (1996) Zbl0862.05032MR1401911DOI10.1006/eujc.1996.0048
  16. Pradhan, D., 10.1016/j.ipl.2012.07.010, Inf. Process. Lett. 112 (2012), 816-822. (2012) Zbl1248.68222MR2960327DOI10.1016/j.ipl.2012.07.010
  17. Tokunaga, S., 10.1016/j.dam.2013.06.025, Discrete Appl. Math. 161 (2013), 3097-3099. (2013) Zbl1287.05109MR3126677DOI10.1016/j.dam.2013.06.025
  18. Tokunaga, S., 10.2197/ipsjjip.28.846, J. Inf. Process. 28 (2020), 846-848. (2020) DOI10.2197/ipsjjip.28.846

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