Counterexamples for Topological Complexity in Digital Images

TL;DR

This paper explores how digital images differ from classical topological spaces in terms of topological complexity, providing counterexamples and relating digital topological complexity to the genus of digital surfaces.

Contribution

It introduces counterexamples highlighting differences between digital images and classical topology, and relates digital topological complexity to digital surface genus.

Findings

Counterexamples show differences in topological complexity between digital images and classical spaces.

Digital topological complexity can be estimated using the genus of digital surfaces.

Results indicate alterations in topological complexities when transitioning from classical to digital topology.

Abstract

Digital topology has its own working conditions and sometimes differs from the normal topology. In the area of topological robotics, we have important counterexamples in this study to emphasize this red line between a digital image and a topological space. We indicate that the results on topological complexities of certain path-connected topological spaces show alterations in digital images. We also give a result about the digital topological complexity number using the genus of a digital surface in discrete geometry.