Independence Complex of the Lexicographic Product of a Forest

We study the independence complex of the lexicographic product $G[H]$ of a forest $G$ and a graph $H$. We prove that for a forest $G$ which is not dominated by a single vertex, if the independence complex of $H$ is homotopy equivalent to a wedge sum of spheres, then so is the independence complex of $G[H]$. We offer two examples of explicit calculations. As the first example, we determine the homotopy type of the independence complex of $L_m [H]$, where $L_m$ is the tree on $m$ vertices with no branches, for any positive integer $m$ when the independence complex of $H$ is homotopy equivalent to a wedge sum of $n$ copies of $d$-dimensional sphere. As the second one, for a forest $G$ and a complete graph $K$, we describe the homological connectivity of the independence complex of $G[K]$ by the independent domination number of $G$.