▁which ▁places ▁the ▁problem ▁in ▁the ▁complexity ▁class ▁G ap P , ▁the ▁closure ▁of ▁# P ▁under ▁sub tra ction . ▁To ▁accommod ate ▁rational ▁coordinates ▁, ▁one ▁can ▁define ▁a ▁rational ▁analog ue ▁of ▁# P . ▁ ▁The ▁computational ▁complexity ▁of ▁exactly ▁computing ▁ ▁falls ▁into ▁one ▁of ▁two ▁classes ▁for ▁any ▁. ▁The ▁problem ▁is ▁# P - hard ▁unless ▁ ▁lies ▁on ▁the ▁hyper bol a ▁ ▁or ▁is ▁one ▁of ▁the ▁points ▁ ▁in ▁which ▁cases ▁it ▁is ▁comput able ▁in ▁polynomial ▁time . ▁If ▁the ▁problem ▁is ▁restricted ▁to ▁the ▁class ▁of ▁plan ar ▁graphs , ▁the ▁points ▁on ▁the ▁hyper bol a ▁ ▁become ▁polynomial - time ▁comput able ▁as ▁well . ▁All ▁other ▁points ▁remain ▁# P - hard , ▁even ▁for ▁b ipart ite ▁plan ar ▁graphs . ▁In ▁his ▁paper ▁on ▁the ▁dich ot omy ▁for ▁plan ar ▁graphs , ▁Vert igan ▁claims ▁( in ▁his ▁conclusion ) ▁that ▁the ▁same ▁result ▁holds ▁when ▁further ▁restricted ▁to ▁graphs ▁with ▁vertex ▁degree ▁at ▁most ▁three , ▁save ▁for ▁the ▁point ▁, ▁which ▁counts ▁now here - zero ▁Z 3 - flow s ▁and ▁is ▁comput able ▁in ▁polynomial ▁time . ▁ ▁These ▁results ▁contain ▁several ▁notable ▁special ▁cases . ▁For ▁example , ▁the ▁problem ▁of ▁computing ▁the ▁partition ▁function ▁of ▁the ▁Is ing ▁model ▁is ▁# P - hard ▁in ▁general , ▁even ▁though ▁celebrated ▁algorithms ▁of ▁On s ager ▁and ▁Fish er ▁solve ▁it ▁for ▁plan ar ▁l att ices . ▁Also