A generating theorem of simple even triangulations with a finitizable set of reductions

We shall determine exactly two (P,Q)-irreducible even triangulations of the projective plane. This result is a new generating theorem of even triangulations of the projective plane, that is, every even triangulation of the projective plane can be obtained from one of those two (P,Q)-irreducible even triangulations by a sequence of two expansions called a P-expansion and a Q-expansion, which were used in Batagelj (1984, 1989), Drapal and Lisonek (2010). Furthermore, we prove that for any closed surface F2 there are finitely many (P,Q)-irreducible even triangulations of F2.