On some multigraph decomposition problems and their computational complexity Original Research Article

Let H be a fixed simple graph. The H-decomposition computational problem is defined as follows: Given an input graph G, can its edge set be partitioned into isomorphic copies of H? The complexity status of H-decomposition problems, where no parallel edges or loops are allowed in G or in H, has been thoroughly studied during the last 20 years and is now completely settled. The subject of this article is the complexity of multigraph decomposition, that is the case where multiple edges are allowed. Apparently, the results obtained here are not always what one would expect by observing the analogous results on simple graphs. For example, deciding whether an input graph G, with fixed multiplicity λ on all edges, can be decomposed into connected subgraphs, each consisting of two distinct edges with multiplicities 1 on one edge and 2 on the other, is NP-complete if λ=2 or 5 and it is solvable in polynomial time for any other values of λ.