NP-Completeness of graph decomposition problems - Cohen, Edith; Tarsi, Michael (Journal of Complexity, 1991)
The H-decomposition problem for a fixed graph H is stated as follows: Can an input graph G be represented as an edge disjoint union of subgraphs, all of which are isomorphic to H? Although H-decomposition problems have been the subject of extensive mathematical research for many decades, even the complexity status of such problems is yet unknown, except for a few families of graphs. H. I. Holyer conjectured that H-decomposition is NP-complete whenever H is connected and has at least 3 edges. The above was proved, however, only for a limited class of graphs H: complete graphs, simple paths, and simple circuits. Holyer's conjecture is proved here for a large family of graphs which contains all trees.