The complexity of decomposing a graph into a matching and a bounded linear forest
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Deciding whether a graph can be edge-decomposed into a matching and a k-bounded linear forest was recently shown by Campbell, Hörsch and Moore to be NP-complete for every k ≥ 9, and solvable in polynomial time for k=1,2. In the first part of this paper, we close this gap by showing that this problem is in NP-complete for every k ≥ 3. In the second part of the paper, we show that deciding whether a graph can be edge-decomposed into a matching and a k-bounded star forest is polynomially solvable for any k ∈ℕ∪{∞}, answering another question by Campbell, Hörsch and Moore from the same paper.
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