Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4

TL;DR

This work investigates the flip graphs of pseudo-triangulations with faces of size at most $4$ (4-PPTs), focusing on the smallest nontrivial case $k=1$ and on the double-chain configuration. The authors develop geometric and combinatorial techniques, including the $dart$ triangle and tail/move lemmas, to analyze connectivity and to construct flip sequences, and they introduce canonical left-aligned forms for the double chain to obtain exact component counts. They prove that the $1$-$DPT$ flip graph is not generally connected and provide an exact method to compute its components, while on the double chain they achieve a complete characterization and a closed-form component count $ ext{min}\\{a,b,k,a+b-k\\}+1$ for $0\le k\le a+b$. These results deepen understanding of 4-PPT flip graphs and offer a foundation for tackling the broader open problem of connectivity in combinatorial $4$-PPT flip graphs, potentially extending to $k\in\{2,3\}$ with further work.

Abstract

A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart pseudo-triangulations ($k$-DPTs), that is, pseudo-triangulations in which exactly $k$ faces are darts and all other faces are triangles. We study the flip graph for such pseudo-triangulations, in which a flip exchanges the diagonals of a pseudo-quadrilatral. Our results are as follows. We prove that the flip graph of $1$-DPTs is generally not connected, and show how to compute its connected components. Furthermore, for $k$-DPTs on a point configuration called the double chain we analyze the structure of the flip graph on a more fine-grained level.

Figures (6)

• Figure 1: Some flips in pseudo-triangulations.
• Figure 2: (a) A dart with tip $a$, tail $b$, wings $c$ and $d$, and a dashed spine. (b) A $1$-DPT on 7 points. (c) An $(n-h)$-DPT with $n=7$ and $h=3$; each vertex not on the convex hull is a dart tail.
• Figure 3: (a) A dart triangle. (b) A dart with an edge connecting the wings. The vertices in the bottom face are split by the extension of the (dashed) spine. (c) The specific triangulation between the dart and the upper envelop of the subset $P'$ of points of $P$ in the triangle formed by the wings and tail of $d$, together with the wings of $d$. (d) A number of flips linear in $|P'|$ creates a dart triangle.
• Figure 4: A flip sequence to flip an edge incident with the face containing the tail $p_d$ of a dart.
• Figure 5: (a)-(d) All dart configurations of five points that allow us to move the tail of a dart using an edge flip. The darts share the blue triangle and each require one dashed edge. (e) Other triangles cannot use both middle vertices as tails, without a (non-existing) point in the yellow area.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 10