On contractible edges in convex decompositions

TL;DR

This paper investigates the properties of convex decompositions of planar point sets, showing that such decompositions always contain either a deletable or a contractible edge, which has implications for geometric graph simplification.

Contribution

It establishes a fundamental property of convex decompositions, proving the existence of either a deletable or a contractible edge in such structures.

Findings

Convex decompositions with more than one polygon always contain a deletable or a contractible edge.

This result provides a basis for simplifying convex decompositions by edge removal or contraction.

The theorem applies to general position point sets in the plane with at least three points.

Abstract

Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.