Triangulations with few ears: symmetry classes and disjointness

TL;DR

This paper studies triangulations of convex polygons with few ears, analyzing their symmetry classes and disjointness properties, and provides explicit counts for cases with two or three ears.

Contribution

It determines the number of symmetry classes and disjoint triangulations for polygons with two or three ears, revealing dependence on polygon size and branch lengths.

Findings

Number of symmetry classes for 2 and 3 ears

Count of disjoint triangulations depending on n or branch lengths

Structural insights into triangulations with few ears

Abstract

An ear in a triangulation $T$ of a convex $n$-gon $P$ is a triangle of $T$ that shares two sides with $P$ itself. Certain enumerational and structural problems become easier when one considers only triangulations with few ears. We demonstrate this in two ways. First, for $k=2, 3$, we find the number of symmetry classes of triangulations with $k$ ears. Second, for $k=2, 3$, we determine the number of triangulations disjoint from a given triangulation: this number depends only on $n$ for $k=2$, and only on lengths of branches of the dual tree for $k=3$.