the G-matrix type $YN$ interaction (the YNG interaction) into the density of the $x$ cluster in the same manner of our previous work on the double-$\Lambda$ hypernuclei [@Hiyama97]. The YNG interactions between $\Lambda$ and $N$ are derived from the $YN$ OBE models as follows: First the G-matrix equation is solved in nuclear matter at each $k_{\rm F}$, where the so called QTQ prescription is adopted for simplicity. Next the resulting G-matrix is simulated by a three-range Gaussian form with the strengths as a function of $k_{\rm F}$. Obtained YNG interactions are given in Ref.[@Yama94] as $$\begin{aligned}
              v_{\Lambda N}\!\!\!&(&\!\!\!\!r ; k_F)  =  \sum_{i=1}^{3}
             \Big[(v_{0,\rm even}^{(i)} + 
            v_{\sigma \sigma,\rm even}^{(i)}
              \mbox{\boldmath $\sigma$}_{\Lambda}\cdot
               \mbox{\boldmath $\sigma$}_N )
               \frac{1+P_r}{2}
                 \nonumber \\
             &+& (v_{0,\rm odd}^{(i)}
                + v_{\sigma \sigma,\rm odd}^{(i)}
                \mbox{\boldmath $\sigma$}_{\Lambda}\cdot
               \mbox{\boldmath $\sigma$}_N )
               \frac{1-P_r}{2} \Big] 
             \: e^{-\mu_i r^2}  , %\nonumber \\
%(3.3)\end{aligned}$$ where $P_r$ is the space exchange (Majorana) operator. The strengths $v^{(i)}$ are represented as quadratic functions of $k_{\rm F}$ ; see Eq.(2.7) of Ref. [@Yama94] and Table V of Ref. [@Hiyama97] for various original $YN$ interactions. In the present work, we employ the Nijmegen model D interaction (ND).

The $\Lambda x$ interaction is derived by folding the above $v_{\Lambda N}(r; k_F)$ interaction into 