  Let $G$ be a graph and $\gamma (G)$ denote the domination number of $G$, i.e.
the cardinality of a smallest set of vertices $S$ such that every vertex of $G$
is either in $S$ or adjacent to a vertex in $S$. Matheson and Tarjan
conjectured that a plane triangulation with a sufficiently large number $n$ of
vertices has $\gamma(G)\le n/4$. Their conjecture remains unsettled. In the
present paper, we show that: (1) a maximal outerplane graph with $n$ vertices
has $\gamma(G)\le \lceil \frac{n+k} 4\rceil$ where $k$ is the number of pairs
of consecutive degree 2 vertices separated by distance at least 3 on the
boundary of $G$; and (2) a Hamiltonian plane triangulation $G$ with $n \ge 23$
vertices has $\gamma (G)\le 5n/16 $. We also point out and provide
counterexamples for several recent published results of Li et al in [Discrete
Appl. Math.198 (2016) 164-169] on this topic which are incorrect.
