  The satisfiability problem is known to be $\mathbf{NP}$-complete in general
and for many restricted cases. One way to restrict instances of $k$-SAT is to
limit the number of times a variable can be occurred. It was shown that for an
instance of 4-SAT with the property that every variable appears in exactly 4
clauses (2 times negated and 2 times not negated), determining whether there is
an assignment for variables such that every clause contains exactly two true
variables and two false variables is $\mathbf{NP}$-complete. In this work, we
show that deciding the satisfiability of 3-SAT with the property that every
variable appears in exactly four clauses (two times negated and two times not
negated), and each clause contains at least two distinct variables is $
\mathbf{NP} $-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular
graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math.,
160(15):2142--2146, 2012] to determine whether for a given independent set $T $
there is an independent dominating set $D$ that dominates $T$ such that $ T
\cap D =\varnothing $? As an application of $(2/2/3)$-SAT problem we show that
for every $r\geq 3$, this problem is $ \mathbf{NP} $-complete. Among other
results, we study the relationship between 1-perfect codes and the incidence
coloring of graphs and as another application of our complexity results, we
prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence
colorable is $ \mathbf{NP} $-complete.
