  A dominating set of a graph $G$ is a subset $S$ of its vertices such that
each vertex of $G$ not in $S$ has a neighbor in $S$. A face-hitting set of a
plane graph $G$ is a set $T$ of vertices in $G$ such that every face of $G$
contains at least one vertex of $T$. We show that the vertex-set of every plane
(multi-)graph without isolated vertices, self-loops or $2$-faces can be
partitioned into two disjoint sets so that both the sets are dominating and
face-hitting. We also show that all the three assumptions above are necessary
for the conclusion.
  As a corollary, we show that every $n$-vertex simple plane triangulation has
a dominating set of size at most $(1 - \alpha)n/2$, where $\alpha n$ is the
maximum size of an independent set in the triangulation. Matheson and Tarjan
[European J. Combin., 1996] conjectured that every plane triangulation with a
sufficiently large number of vertices $n$ has a dominating set of size at most
$n / 4$. Currently, the best known general bound for this is by Christiansen,
Rotenberg and Rutschmann [SODA, 2024] who showed that every plane triangulation
on $n > 10$ vertices has a dominating set of size at most $2n/7$. Our corollary
improves their bound for $n$-vertex plane triangulations which contain a
maximal independent set of size either less than $2n/7$ or more than $3n/7$.
