Efficient algorithms for decomposing graphs under degree constraints

Abstract

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321–324] 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 $d_A(v)⩾a(v)$ for every $v\in A$ and $d_B(v)⩾b(v)$ for every $v\in B$. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7–9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237–239] 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 polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.