Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete

  title={Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete},
  author={Aviezri S. Fraenkel},
  journal={Discrete Applied Mathematics},
It is proved that the questions whether a finite digraph G has a kernel K or a Sprague-Grundy function g are NP-complete even if G is a cyclic planar digraph with degree constraints dOut(u)s 2, d,,(u) ~2 and d(u) 2 3. These results are best possible (if P f NP) in the sense that if any of the constraints is tightened, there are polynomial algorithms which either compute K and g or show that they do not exist. The proof uses a single reduction from planar 3.satisfiability for both problems. 

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