Bridging Graph-Theoretical and Topological Approaches: Connectivity and Jordan Curves in the Digital Plane

TL;DR

The results illustrate the deep and intricate relationship between the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.

Abstract

This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on $\mathbb{Z}^2$, and the subsequent introduction of the topological digital plane $\mathbb{K}^2$ with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator $Γ^*$ transforming subsets of $\mathbb{Z}^2$ into subsets of $\mathbb{K}^2$. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in $\mathbb{Z}^2$ correspond to topological properties in $\mathbb{K}^2$. Moreover, we address whether the topological Jordan curve theorem for $\mathbb{K}^2$ can be derived from the graph-theoretical version on $\mathbb{Z}^2$. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.

Figures (12)

• Figure 1: $N_4(p)$ and $N_8(p)$.
• Figure 2: Labeling of the 8-neighborhood points of $p$.
• Figure 3: $p_2$ and $p_4$ are in different 4-components.
• Figure 4: Adjacency in the digital plane $\mathbb{K}^2$.
• Figure 5: Rosenfeld plane embedded in the digital plane.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Theorem 2.4
• Proposition 2.5
• proof
• Definition 3.1
• Theorem 3.2