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arXiv · 2011.04255

Total domination in plane triangulations

M. Claverol·A. García·G. Hernández·C. Hernando·M. Maureso·M. Mora·J. Tejel
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Abstract

A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $\gamma _t (G)$, is the minimum cardinality of a total dominating set of $G$. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that $\gamma _t (G) \le \lfloor \frac{2n}{5}\rfloor$ for any near-triangulation $G$ of order $n\ge 5$, with two exceptions.

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M. Claverol, A. García, G. Hernández, C. Hernando, M. Maureso, M. Mora, J. Tejel. 2020-11-09. Total domination in plane triangulations. https://doi.org/10.1016/j.disc.2020.112179

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