We continue research into a well-studied family of problems that ask whether the vertices of a given graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class . We consider the case where  is the class of k-degenerate graphs. This problem is known to be polynomial-time solvable if k = 0 (recognition of bipartite graphs), but NPcomplete if k = 1 (near-bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan

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... that the k = 1 case is polynomial-time solvable for graphs of maximum degree 3. This also follows from a result of Catlin and Lai. We study the general ≥ k 1 case for n-vertex graphs of maximum degree k + 2. We show how to find A and B in O n ( ) time for k = 1, and in O n ( ) 2 time for ≥ k 2. Together, these results provide an algorithmic version of a result of Catlin and also provide an algorithmic version of a generalization of Brook's Theorem, proved by Borodin et al. and Matamala. The results also enable us to solve an open problem of Feghali et al. For a given graph G and positive integer ℓ, the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.