  We investigate the relationship between the univalence of $f$ and of $h$ in
the decomposition $f=h+\bar{g}$ of a sense-preserving harmonic mapping defined
in the unit disk $\mathbb{D}\subset\mathbb{C}$. Among other results, we
determine the holomorphic univalent maps $h$ for which there exists $c>0$ such
that every harmonic mapping of the form $f=h+\bar{g}$ with $|g'|< c|h'|$ is
univalent. The notion of a linearly connected domain appears in our study in a
relevant way.
