Trimming and Building Freezing Sets

TL;DR

This paper introduces new methods for constructing fixed point sets in digital topology by identifying excludable points, including articulation points, and characterizing points that must be included in freezing sets.

Contribution

It presents novel tools for constructing freezing sets, including the concept of excludable points and criteria for points that must be included.

Findings

Articulation points are excludable from freezing sets.

Points of degree 1 and local extrema often must be included in freezing sets.

New methods simplify the construction of fixed point sets in digital topology.

Abstract

We develop new tools for the construction of fixed point sets in digital topology. We define excludable points and show that these may be excluded from all freezing sets. We show that articulation points are excludable. We also present results concerning points that must belong to a freezing set and often are easily recognized. These include points of degree~1 and some local extrema.