Polynomial cases of graph decomposition: A complete solution of Holyer’s problem

Let H be a fixed graph. A graph G has an H -decomposition if the edge set of G can be partitioned into subsets inducing graphs isomorphic to H . Let P H denote the following decision problem: “Does an instance graph G admit H -decomposition?” In this paper we prove that the problem P H is polynomial time solvable if H is a graph whose every component has at most 2 edges. This way we complete a solution of Holyer’s problem which is the problem of classifying the problems P H according to their computational complexities.