The problem of determining if a graph is 2-colourable (i.e., bipartite) has long been known to have a simple polynomial time algorithm. Being 2-colourable is equivalent to having a bipartition of the vertex set where each cell is K2-free. We extend this notion to determining if there exists a bipartition where each cell is G-free for some fixed graph G. One might expect that for some graphs other than there also exist polynomial time algorithms. Rather surprisingly we show that for any graph G on more than two vertices the problem is NP-complete.
