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 \(G\).