Decomposition of Multigraphs into Isomorphic Graphs with Two Edges

We improve upon Caro’s general polynomial characterizations, all in terms of modified line graphs, restricted to decomposing a graph into isomorphic subgraphs \(H\) with two edges. Firstly, we solve the problem for a multigraph; secondly, we decrease the polynomial bound on complexity if \(H = 2K_2\) and provide an original sufficient condition which can be verified in linear time if \(H = P_3\).