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 $F^2$ there are finitely many $(P,Q)$-irreducible even triangulations of $F^2$.