  In a phenomenological approach, the $\alpha$ + core structure is investigated
in the $^{20}$Ne, $^{44}$Ti, $^{94}$Mo, $^{104}$Te, and $^{212}$Po nuclei
through the local potential model using a double-folding nuclear potential with
effective nucleon-nucleon interaction of M3Y + $c_{\mathrm{sat}}\delta(s)$
type, where the term $c_{\mathrm{sat}}\delta(s)$ acts only between the
saturation regions of $\alpha$-cluster and core. Properties such as energy
levels, $\alpha$-widths, $B(E2)$ transition rates, and half-lives are
calculated, and good level of agreement with experimental data is obtained in
general. It is shown the inclusion of the term $c_{\mathrm{sat}}\delta(s)$ is
determinant for a better description of the experimental energy levels compared
to the ground state bands produced with the simple M3Y interaction. The
properties calculated for $^{104}$Te reinforce the superallowed $\alpha$-decay
feature for this nucleus, as indicated in previous studies.
