An H-decomposition of a graph G=(V,E) is a partition of E into subgraphs isomorphic to H. Given a fixed graph H, the H-decomposition problem is to determine whether an input graph G admits an H-decomposition. In 1980, Holyer conjectured that H-decomposition is NP-complete whenever H is connected and has three edges or more. Some partial results have been obtained since then. A complete proof of Holyer's conjecture is the content of this paper. The characterization problem of all graphs H for which H-decomposition is NP-complete is hence reduced to graphs where every connected component contains at most two edges.

An H-decomposition of a graph G = (V,E) is a partition of E into subgraphs isomorphic to H. Given a fixed graph H, the H-decomposition problem is to determine whether an input graph G admits an H-decomposition.

It is proved that the dlrecUonal algorithm for solving a game tree is optimal, in the sense of average run trine, for balanced trees (a family containing all uniform trees).This result implies that the alpha-beta pruning method is asymptotJcally opttmal among all game searching algorithms. Categones and Subject

We find the optimal algorithm in the sense of average run time for the counterfeit coin problem: Given n coins, one of which is heavier or lighter than the rest. Using a balance scale, find the counterfeit coin and whether it is heavy or light. An interesting feature of the solution is that our algorithm is a straight line algorithm. We also find the optimal algorithm if a standard coin is available for the first weighing.