On irreducible triangulations of punctured and pinched surfaces

Abstract

A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible triangulations are determined for the Möbius band (6 in number) and the pinched torus (2 in number). All the non-isomorphic combinatorial types (20 in number) of triangulations of the projective plane with up to 8 vertices are determined.

This paper has been withdrawn by the authors because the proof of Lemma 3.3 has a gap. More precisely, the claim "If R has a pylonic vertex, v, incident with at least two cables, the pylonicity of v is destroyed by the splitting of any corner", as stated, is unjustified and looks false in whole generality; the authors overlooked some cases