2 research outputs found
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate
Surface realization with the intersection edge functional
Deciding realizability of a given polyhedral map on a (compact, connected)
surface belongs to the hard problems in discrete geometry, from the
theoretical, the algorithmic, and the practical point of view.
In this paper, we present a heuristic algorithm for the realization of
simplicial maps, based on the intersection edge functional. The heuristic was
used to find geometric realizations in R^3 for all vertex-minimal
triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for
the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12
vertices were obtained.Comment: 22 pages, 11 figures, various minor revisions, to appear in
Experimental Mathematic