Geodesic
On the domination of triangulated discs
Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 555-560
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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 $\frac 14(n+2)$. 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.
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 $\frac 14(n+2)$. 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.
DOI : 10.21136/MB.2022.0122-21
Classification : 05C69
Keywords: domination; double domination; total domination; double total domination; planar graph; triangulated disc 
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Abd Aziz, Noor A'lawiah; Jafari Rad, Nader; Kamarulhaili, Hailiza. On the domination of triangulated discs. Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 555-560. doi: 10.21136/MB.2022.0122-21

[1] Bermudo, S., Sanchéz, J. L., Sigarreta, J. M.: Total $k$-domination in Cartesian product graphs. Period. Math. Hung. 75 (2017), 255-267. | DOI | MR | JFM

[2] Blidia, M., Chellali, M., Haynes, T. W.: Characterizations of trees with equal paired and double domination numbers. Discrete Math. 306 (2006), 1840-1845. | DOI | MR | JFM

[3] Cabrera-Martínez, A.: New bounds on the double domination number of trees. Discrete Appl. Math. 315 (2022), 97-103. | DOI | MR | JFM

[4] Cabrera-Martínez, A., Rodríguez-Velázquez, J. A.: A note on double domination in graphs. Discrete Appl. Math. 300 (2021), 107-111. | DOI | MR | JFM

[5] Campos, C. N., Wakabayashi, Y.: On dominating sets of maximal outerplanar graphs. Discrete Appl. Math. 161 (2013), 330-335. | DOI | MR | JFM

[6] Fernau, H., Rodríguez-Velázquez, J. A., Sigarreta, J. M.: Global powerful $r$-alliances and total $k$-domination in graphs. Util. Math. 98 (2015), 127-147. | MR | JFM

[7] Harant, J., Henning, M. A.: On double domination in graphs. Discuss. Math., Graph Theory 25 (2005), 29-34. | DOI | MR | JFM

[8] Harary, F., Haynes, T. W.: Double domination in graphs. Ars Comb. 55 (2000), 201-213. | MR | JFM

[9] Haynes, T. W., Hedetniemi, S. T., Slater, P. J.: Fundamentals of Domination in Graphs. Pure and Applied Mathematics, Marcel Dekker 208. Marcel Dekker, New York (1998). | DOI | MR | JFM

[10] Henning, M. A., Rad, N. Jafari: Upper bounds on the $k$-tuple (Roman) domination number of a graph. Graphs Comb. 37 (2021), 325-336. | DOI | MR | JFM

[11] Henning, M. A., Kazemi, A. P.: $k$-tuple total domination in graphs. Discrete Appl. Math. 158 (2010), 1006-1011. | DOI | MR | JFM

[12] Henning, M. A., Yeo, A.: Strong transversals in hypergraphs and double total domination in graphs. SIAM J. Discrete Math. 24 (2010), 1336-1355. | DOI | MR | JFM

[13] King, E. L. C., Pelsmajer, M. J.: Dominating sets in plane triangulations. Discrete Math. 310 (2010), 2221-2230. | DOI | MR | JFM

[14] Li, Z., Zhu, E., Shao, Z., Xu, J.: On dominating sets of maximal outerplanar and planar graphs. Discrete Appl. Math. 198 (2016), 164-169. | DOI | MR | JFM

[15] Matheson, L. R., Tarjan, R. E.: Dominating sets in planar graphs. Eur. J. Comb. 17 (1996), 565-568. | DOI | MR | JFM

[16] Pradhan, D.: Algorithmic aspects of $k$-tuple total domination in graphs. Inf. Process. Lett. 112 (2012), 816-822. | DOI | MR | JFM

[17] Tokunaga, S.: Dominating sets of maximal outerplanar graphs. Discrete Appl. Math. 161 (2013), 3097-3099. | DOI | MR | JFM

[18] Tokunaga, S.: On domination number of triangulated discs. J. Inf. Process. 28 (2020), 846-848. | DOI

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