Noor A'lawiah Abd Aziz, Nader Jafari Rad, Hailiza Kamarulhaili
Received August 10, 2021. Published online December 5, 2022.
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 $\frac14(n+2)$. 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.
Keywords: domination; double domination; total domination; double total domination; planar graph; triangulated disc
Affiliations: Noor A'lawiah Abd Aziz, School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia, e-mail: email@example.com; Nader Jafari Rad (corresponding author), Department of Mathematics, Shahed University, Tehran, Iran, e-mail: firstname.lastname@example.org; Hailiza Kamarulhaili, School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia, e-mail: email@example.com