# Eccentricities in the flip-graphs of convex polygons

## TL;DR

This paper generalizes the understanding of eccentricities in flip-graphs of convex polygons, relating triangulation properties to their maximum distances within the graph, and provides bounds and characterizations for these eccentricities.

## Contribution

It extends known results by characterizing eccentricities for arbitrary triangulations based on interior edges incident to a vertex, and establishes bounds and inverse relations.

## Key findings

- Eccentricity of triangulations is exactly $n-3+k$ under certain conditions.
- Small eccentricities uniquely determine the number of interior edges incident to a vertex.
- Bounds on eccentricity are provided for larger values of $k$.

## Abstract

The flip-graph of a convex polygon $\pi$ is the graph whose vertices are the triangulations of $\pi$ and whose edges correspond to flips between them. The eccentricity of a triangulation $T$ of $\pi$ is the largest possible distance in this graph from $T$ to any triangulation of $\pi$. It is well known that, when all $n-3$ interior edges of $T$ are incident to the same vertex, the eccentricity of $T$ in the flip-graph of $\pi$ is exactly $n-3$, where $n$ denotes the number of vertices of $\pi$. Here, this statement is generalized to arbitrary triangulations. Denoting by $n-3-k$ the largest number of interior edges of $T$ incident to a vertex, it is shown that the eccentricity of $T$ in the flip-graph of $\pi$ is exactly $n-3+k$, provided $k\leq{n/2-2}$. Inversely, the eccentricity of a triangulation, when small enough, allows to recover the value of $k$. More precisely, if $k\leq{n/8-5/2}$, it is also shown that $T$ has eccentricity $n-3+k$ if and only if exactly $n-3-k$ of its interior edges are incident to a given vertex. When $k>n/2-2$, bounds on the eccentricity of $T$ are also given and discussed.