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On the Connectivity of the Flip Graph of Plane Spanning Paths

Linda Kleist, Peter Kramer, Christian Rieck

TL;DR

This work tackles the open problem of connectivity for the flip graph of plane spanning paths on general-position point sets by focusing on strategically tractable subgraphs. It develops a framework based on suffix-independence and layer-monotone structure, proving that the suffix-independent subgraph $\mathcal{P}_{\sin}(S,s)$ is connected and establishing tight diameter and radius for convex-position sets with a fixed start $s$ (diameter $$2n-5$$, radius $n-2$), with spirals at the centers. It then shows the flip graph is connected for point sets with at most two convex layers by combining suffix-independent connectivity with a two-layer property, providing concrete progress toward the general conjecture. The results introduce tools such as layer-based decompositions and constructive flip sequences that reduce global reconfiguration questions to suffix-independent reachability, with potential implications for planning and optimization in planar non-crossing configurations.

Abstract

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a flip graph. In stark contrast, the connectivity of the flip graph of plane spanning paths on point sets in general position has been an open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs. Firstly, we provide tight bounds on the diameter and the radius of the flip graph of spanning paths on points in convex position with one fixed endpoint. Secondly, we show that so-called suffix-independent paths induce a connected subgraph. Consequently, to answer the open problem affirmatively, it suffices to show that each path can be flipped to some suffix-independent path. Lastly, we investigate paths where one endpoint is fixed and provide tools to flip to suffix-independent paths. We show that these tools are strong enough to show connectivity of the flip graph of plane spanning paths on point sets with at most two convex layers.

On the Connectivity of the Flip Graph of Plane Spanning Paths

TL;DR

This work tackles the open problem of connectivity for the flip graph of plane spanning paths on general-position point sets by focusing on strategically tractable subgraphs. It develops a framework based on suffix-independence and layer-monotone structure, proving that the suffix-independent subgraph is connected and establishing tight diameter and radius for convex-position sets with a fixed start (diameter , radius ), with spirals at the centers. It then shows the flip graph is connected for point sets with at most two convex layers by combining suffix-independent connectivity with a two-layer property, providing concrete progress toward the general conjecture. The results introduce tools such as layer-based decompositions and constructive flip sequences that reduce global reconfiguration questions to suffix-independent reachability, with potential implications for planning and optimization in planar non-crossing configurations.

Abstract

Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a flip graph. In stark contrast, the connectivity of the flip graph of plane spanning paths on point sets in general position has been an open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs. Firstly, we provide tight bounds on the diameter and the radius of the flip graph of spanning paths on points in convex position with one fixed endpoint. Secondly, we show that so-called suffix-independent paths induce a connected subgraph. Consequently, to answer the open problem affirmatively, it suffices to show that each path can be flipped to some suffix-independent path. Lastly, we investigate paths where one endpoint is fixed and provide tools to flip to suffix-independent paths. We show that these tools are strong enough to show connectivity of the flip graph of plane spanning paths on point sets with at most two convex layers.
Paper Structure (7 sections, 15 theorems, 9 figures)

This paper contains 7 sections, 15 theorems, 9 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Fundamental concepts and tools
  5. Plane paths on point sets in convex position
  6. The subgraph of suffix-independent paths is connected
  7. General tools and point sets on two convex layers

Key Result

Theorem 3

Let $S$ be a set of $n$ points in convex position and let $s\in S$. For ${n\geq3}$, the flip graph of $\mathcal{P}(S,s)$ has diameter $2n-5$ and radius $n-2$. Moreover, the spirals are exactly the centers of the flip graph.

Figures (9)

  • Figure 1: Plane spanning paths that can be transformed into each other by a flip.
  • Figure 2: The flip graph of plane spanning paths with a fixed start on a point set with two layers, where each layer contains three points. Six paths, highlighted in orange, are not suffix-independent. The two spirals, highlighted in red, have flip distance 5 and are a furthest pair.
  • Figure 3: A flip on a path with a fixed endpoint, illustrating \ref{['lem:fixed-flips']}.
  • Figure 4: A point set with (a) its layers, (b) a chord, an inward cutting edge, and a level edge (from left to right), and (c) its counterclockwise spiral from $s\in L_0$.
  • Figure 5: A pair of points divides each layer into $L_0^+(u,v;w)$ and $L_0^-(u,v;w)$.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Conjecture 1: Akl et al. akl2007planar
  • Conjecture 2: Aichholzer et al. 2023Aicholzer
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Theorem 10
  • ...and 7 more