# Minimal 3-triangulations of $p$-toroid

## TL;DR

This paper investigates the minimal number of tetrahedra needed to 3-triangulate p-toroids, a class of non-convex polyhedra topologically equivalent to spheres with p handles, using new concepts like piecewise convexity.

## Contribution

It introduces the concepts of piecewise convex polyhedra and graphs of connection to analyze 3-triangulations of p-toroids and determines minimal tetrahedral counts.

## Key findings

- Established lower bounds for tetrahedral counts in p-toroids
- Developed new theoretical tools for analyzing non-convex polyhedra
- Extended triangulation theory to complex topologies

## Abstract

It is known that we can always 3-triangulate (i.e. divide into tetrahedra) convex polyhedra but not always non-convex ones. Polyhedra topologically equivalent to sphere with $p$ handles, shortly $p$-toroids, could not be convex. So, it is interesting to investigate possibilities and properties of their 3-triangulations. Here, we study the minimal necessary number of tetrahedra for the triangulation of a 3-triangulable $p$-toroid. For that purpose, we developed the concepts of piecewise convex polyhedra and graphs of connection.