On the domination of 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 $\frac14(n+2)$. This conjecture is proved in Tokunaga (2020) for $G-C$ being a tree. In this paper we prove...

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Published inMathematica bohemica Vol. 148; no. 4; pp. 555 - 560
Main Authors Noor A'lawiah Abd Aziz, Nader Jafari Rad, Hailiza Kamarulhaili
Format Journal Article
LanguageEnglish
Published Institute of Mathematics of the Czech Academy of Science 01.12.2023
Subjects
domination
double domination
double total domination
planar graph
total domination
triangulated disc
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Summary: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.
ISSN:0862-7959
2464-7136
DOI:10.21136/MB.2022.0122-21