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We consider a certain class of simplicial complexes which includes theindependence complexes of forests. We show that if a simplicial complex $K$belongs to this class, then the polyhedral join$\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedgesum of CW complexes of the form $\Sigma^r X_{i_1} * X_{i_2} * \cdots *X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CWcomplexes and $\Sigma$ denotes the unreduced suspension. This result is appliedto study the homotopy type of the independence complex of the lexicographicproduct $G[H]$ of a graph $H$ over a forest $G$. We denote by $L_m$ a tree on$m$ vertices with no branches. We show that the geometric realization of theindependence complex of $L_m [H]$ is homotopy equivalent to a wedge sum ofspheres if $m \neq 2,3$ and the geometric realization of the independencecomplex of $H$ is homotopy equivalent to a wedge sum of same dimensionalspheres.

On the Homotopy Type of the Polyhedral Join over the Independence Complex of a Forest