Flipping Plane Spanning Paths

02/22/2022
βˆ™
by   Oswin Aichholzer, et al.
βˆ™
0
βˆ™

Let S be a planar point set in general position, and let 𝒫(S) be the set of all plane straight-line paths with vertex set S. A flip on a path P βˆˆπ’«(S) is the operation of replacing an edge e of P with another edge f on S to obtain a new valid path from 𝒫(S). It is a long-standing open question whether for every given planar point set S, every path from 𝒫(S) can be transformed into any other path from 𝒫(S) by a sequence of flips. To achieve a better understanding of this question, we provide positive answers for special classes of point sets, namely, for wheel sets, ice cream cones, double chains, and double circles. Moreover, we show for general point sets, it is sufficient to prove the statement for plane spanning paths whose first edge is fixed.

READ FULL TEXT

page 1

page 3

page 5

page 7

page 9

page 13

page 19

research
βˆ™ 06/12/2023

Three Edge-disjoint Plane Spanning Paths in a Point Set

We study the following problem: Given a set S of n points in the plane, ...
research
βˆ™ 09/13/2019

Linear Size Planar Manhattan Network for Convex Point Sets

Let G = (V, E) be an edge-weighted geometric graph such that every edge ...
research
βˆ™ 03/15/2022

Chains, Koch Chains, and Point Sets with many Triangulations

We introduce the abstract notion of a chain, which is a sequence of n po...
research
βˆ™ 03/08/2023

Improved Bounds for Covering Paths and Trees in the Plane

A covering path for a planar point set is a path drawn in the plane with...
research
βˆ™ 07/20/2020

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a s...
research
βˆ™ 10/29/2019

Equipartitions with Wedges and Cones

A famous result about mass partitions is the so called Ham-Sandwich theo...
research
βˆ™ 09/22/2022

Piercing Diametral Disks Induced by Edges of Maximum Spanning Tree

Let P be a set of points in the plane and let T be a maximum-weight span...

Please sign up or login with your details

Forgot password? Click here to reset