A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection

Abstract

We provide a new technique for deriving optimal sized polygonal schema for triangulated compact 2-manifolds without boundary in O(n) time, where n is the size of the given triangulation T. We first derive a polygonal schema P embedded in T using Seifert-Van Kampen's theorem. A reduced polygonal schema Q of optimal size is computed from P, where a surjective mapping from the vertices of P is retained to the vertices of Q. This helps detecting null-homotopic (contractable to a point) cycles. Given a cycle of length k we determine if it is null-homotopic in O(n+gk) time where g is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in O(gkn) time and space. This is an improvement of a factor of g over the previous best-known algorithms for these problems.