  Motivated by recent observational data, the equations of state with the
inclusion of strangeness-bearing $\Lambda$-hyperons and the corresponding
properties of neutron stars are studied, based on the covariant density
functional (CDF) theory. To this end, we specifically employ the density
dependent relativistic Hartree-Fock (DDRHF) theory and the relativistic mean
field theory (RMF). The inclusion of $\Lambda$-hyperons in neutron stars shows
substantial effects in softening the equation of state. Because of the extra
suppression effect originated from the Fock channel, large reductions on both
the star mass and radius are predicted by the DDRHF calculations. It is also
found that the mass-radius relations of neutron stars with $\Lambda$-hyperons
determined by DDRHF with the PKA1 parameter set are in fairly good agreement
with the observational data where a relatively small neutron stars radius is
required. Therefore, it is expected that the exotic degrees of freedom such as
the strangeness-bearing structure may appear and play significant roles inside
the neutron stars, which is supported further by the systematical
investigations on the consistency between the maximum neutron star mass and
$\Lambda$-coupling strength.
