Jakob Skwarski: Operations on a graph G that shift the homology of the independence complex of G

For a graph G we describe operations on certain kinds of subgraphs of G to get a new graph G' such that I(G';-1) = ±I(G;-1), where I(G;x) is the independence polynomial of G. This is true no matter how the graph G looks outside of the subgraph we study. We show that the independence complexes of G and of G' have similar homology groups. There are known results which state that there exist bipartite graphs whose independence complex contains arbitrarily complicated homology groups. Using the new operations we show a similar result for planar graphs.