Efficient algorithms for decomposing graphs under degree constraints

Stiebitz (1996) proved that if every vertex v in a graph G has degree d(v) ≥ a(v)+ b(v) + 1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that dA(v) ≥ a(v) for every v ∈ A and dB(v) ≥ b(v) for every v ∈ B. Kaneko (1998) and Diwan (2000) strengthened this result, proving that it suffices to assume d(v) ≥ a+b (a, b ≥ 1) or just d(v) ≥ a+b−1 (a, b ≥ 2) if G contains no cycles shorter than 4 or 5, respectively. The original proofs contain nonconstructive steps. In this paper we give polynomialtime algorithms that find such partitions. Constructive generalizations for k-partitions