AbstractA pair of graphs (H1,H2) is a cleaver if there is a graphGwith a unique factorizationG = G2⊕G2such thatGidoes not contain a copy ofHi. (In caseH1=H2,H1is a cleaver if the factorization is unique up to interchange of factors.) A variety of positive and negative results on the existence of cleavers is obtained. For example (tree, complete graph) is a cleaver, and the triangle is the cleaver of an infinite collection of minimal graphs. On the other hand, stars are not cleaver