On independence complexes of graph products

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$ and $P_n\boxtimes P_4$. We also focus in the independence complexes of the induced subgraphs of $P_n\times P_3$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$, $P_n\boxtimes P_4$ and some lexicographic products $G\circ H$.