  For a maximal outerplanar graph $G$ of order $n$ at least $3$, Matheson and
Tarjan showed that $G$ has domination number at most $n/3$. Similarly, for a
maximal outerplanar graph $G$ of order $n$ at least $5$, Dorfling, Hattingh,
and Jonck showed, by a completely different approach, that $G$ has total
domination number at most $2n/5$ unless $G$ is isomorphic to one of two
exceptional graphs of order $12$.
  We present a unified proof of a common generalization of these two results.
For every positive integer $k$, we specify a set ${\cal H}_k$ of graphs of
order at least $4k+4$ and at most $4k^2-2k$ such that every maximal outerplanar
graph $G$ of order $n$ at least $2k+1$ that does not belong to ${\cal H}_k$ has
a dominating set $D$ of order at most $\lfloor\frac{kn}{2k+1}\rfloor$ such that
every component of the subgraph $G[D]$ of $G$ induced by $D$ has order at least
$k$.
