Star Decompositions of a Cyclic Polygon

TL;DR

This work studies star decompositions of cyclic polygons, introducing a framework of cycle decompositions built from diagonals and stars. It proves that any two maximal star decompositions of a common cyclic polygon are connected by a finite sequence of diagonal flips, and establishes a precise count of diagonals in a maximal decomposition as $p - \frac{(n-2r)(n-2r-1)}{2}$, where $p$ is the total number of diagonals, $n$ the vertex count, and $r$ the rotation number. The paper also discusses existence conditions for star decompositions and provides exact counts for special families like $P_n^k$, linking to Catalan-related structures and Dick-path fans. These results extend combinatorial triangulation concepts to generalized star decompositions, with implications for pseudo-triangulations and related geometric complexes.

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 $P$.

Figures (12)

• Figure 1: A cyclic polygon(left) and one of its maximal star decompositions with a set of additional diagonals(right).
• Figure 2: $k$-stars for $k=1$(left), $k=2$(center), $k=3$(right).
• Figure 3: Two stars $S_1 = u_1 u_2 u_3 u_4 u_5 u_1$ and $S_2 = v_1 v_2 v_3 v_1$ have three linkable pairs $(u_1, v_2)$, $(u_4, v_3)$ and $(u_5, v_1)$ (left). By a star subdivision using $(u_1, v_2)$ and $(u_5, v_1)$, we have new stars $S'_1 = u_1 v_2 v_3 v_1 u_5 u_1$ and $S'_2 = v_2 u_1 u_2 u_3 u_4 u_5 v_1 v_2$(right).
• Figure 4: $k$-triangulations for $k=1$(left), $k=2$(center), $k=3$(right).
• Figure 5: The vertices $V_1$ and $V_2$ are displayed in black and white, respectively. Labels are their weights.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 6
• proof
• Lemma 7
• proof