Every Combinatorial Polyhedron Can Unfold with Overlap

12/30/2022
by   Joseph O'Rourke, et al.
0

Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result as establishing that every combinatorial polyhedron has a metric realization that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of “reverse"): Is there a combinatorial polyhedron such that, for every metric realization P in R^3, and for every spanning cut-tree T, P cut by T unfolds to a net? In this note we prove the answer is NO: every combinatorial polyhedron has a realization and a cut-tree that unfolds the polyhedron with overlap.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/02/2018

Lean tree-cut decompositions: obstructions and algorithms

The notion of tree-cut width has been introduced by Wollan [The structur...
research
10/07/2019

Faster Minimum k-cut of a Simple Graph

We consider the (exact, minimum) k-cut problem: given a graph and an int...
research
10/24/2022

Edge-Cuts and Rooted Spanning Trees

We give a closed form formula to determine the size of a k-respecting cu...
research
02/22/2021

Cut Locus Realizations on Convex Polyhedra

We prove that every positively-weighted tree T can be realized as the cu...
research
06/30/2022

Slim Tree-Cut Width

Tree-cut width is a parameter that has been introduced as an attempt to ...
research
07/28/2020

Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra

We present new examples of topologically convex edge-ununfoldable polyhe...
research
01/18/2018

A universality theorem for allowable sequences with applications

Order types are a well known abstraction of combinatorial properties of ...

Please sign up or login with your details

Forgot password? Click here to reset