On the Connectivity of the Flip Graph of Plane Spanning Paths

TL;DR

This paper investigates the connectivity of flip graphs of plane spanning paths, providing bounds, subgraph analyses, and tools that contribute to solving an open problem in computational geometry.

Contribution

It introduces bounds on diameter and radius, analyzes suffix-independent paths, and proves connectivity for certain point set configurations.

Findings

Bounds on diameter and radius for convex position with fixed endpoint

Suffix-independent paths form a connected subgraph

Connectivity established for point sets with up to two convex layers

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.