On the domination of triangulated discs

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