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

@article{Fraenkel1981PlanarKA, title={Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete}, author={Aviezri S. Fraenkel}, journal={Discrete Applied Mathematics}, year={1981}, volume={3}, pages={257-262} }

- Published 1981 in Discrete Applied Mathematics
DOI:10.1016/0166-218X(81)90003-2

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.