  Several domination results have been obtained for maximal outerplanar graphs
(mops). The classical domination problem is to minimize the size of a set $S$
of vertices of an $n$-vertex graph $G$ such that $G - N[S]$, the graph obtained
by deleting the closed neighborhood of $S$, is null. A classical result of
Chv\'{a}tal is that the minimum size is at most $n/3$ if $G$ is a mop. Here we
consider a modification by allowing $G - N[S]$ to have isolated vertices and
isolated edges only. Let $\iota_1(G)$ denote the size of a smallest set $S$ for
which this is achieved. We show that if $G$ is a mop on $n \geq 5$ vertices,
then $\iota_{1}(G) \leq n/5$. We also show that if $n_2$ is the number of
vertices of degree $2$, then $\iota_{1}(G) \leq \frac{n+n_2}{6}$ if $n_2 \leq
\frac{n}{3}$, and $\iota_1(G) \leq \frac{n-n_2}{3}$ otherwise. We show that
these bounds are best possible.
