Star Decompositions of a Cyclic Polygon

TL;DR

This paper studies the structure of star decompositions in cyclic polygons, proving their transformability via diagonal flips and deriving formulas for the number of diagonals in maximal decompositions.

Contribution

It introduces the concept of maximal star decompositions in cyclic polygons and proves their connectivity through diagonal flips, also providing a formula for the number of diagonals.

Findings

Any two maximal star decompositions can be transformed into each other by diagonal flips.

The number of diagonals in a maximal star decomposition is given by a specific formula involving vertices and rotation number.

The paper establishes structural properties of star decompositions in cyclic polygons.

Abstract

Let $V$ be a set of vertices on a circumference in the plane. Let $E$ be a set of directed line segments linking two vertices of $V$. If $E$ forms a set of closed cycles and for all two adjacent edges $uv$ and $vw$, the vertices $u$, $v$, $w$ are arranged in anti-clockwise order, we call $P(V,E)$ a cyclic polygon. A star decomposition $\mathcal{S}$ of a cyclic polygon $P$ is a set of star polygons partitioning the region of $P$ with some additional diagonals. A star decomposition $\mathcal{S}$ is called maximal if there is no other star decomposition $\mathcal{S}'$ such that a set of diagonals of $\mathcal{S}$ is a proper subset of that of $\mathcal{S}'$. In this paper, it is shown that for any two maximal star decompositions $\mathcal{S}_1$ and $\mathcal{S}_2$ of a common cyclic polygon, $\mathcal{S}_1$ can be transformed into $\mathcal{S}_2$ by a finite sequence of diagonal flips. It is also shown that if a cyclic polygon $P$ admits a star decomposition, the number of diagonals contained in a maximal star decomposition of $P$ is $p - (n-2r)(n-2r-1)/2$, where $p$ is the number of all possible diagonals of $P$, $n$ is the number of vertices of $P$, and $r$ is the rotation number of $P$.