Decomposition of graphs: some polynomial cases

We study the problem of decomposing the vertex set V of a graph into two parts (V1, V2) which induce subgraphs where each vertex v in V1 has degree at least a(v) and each vertex v in V2 has degree at least b(v). We investigate several polynomial cases of this NP complete problem. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition and gives such a decomposition if it exists. We also give polynomial-time algorithms that always find a decomposition for the following two cases : triangle-free graphs such that d(v) ≥ a(v) + b(v) for all v ∈ V and graphs with girth at least 5 such that d(v) ≥ a(v) + b(v)− 1 for all v ∈ V .