For every d≥2, d-Cut is constant-time solvable on claw-free graphs of maximum degree at most 2d+1 and NP-complete on those of degree at least 2d+3, leaving only degree 2d+2 open.
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Finding d-Cuts in Claw-free Graphs
For every d≥2, d-Cut is constant-time solvable on claw-free graphs of maximum degree at most 2d+1 and NP-complete on those of degree at least 2d+3, leaving only degree 2d+2 open.