Bridging graph-theoretical and topological approaches: Connectivity and Jordan curves in the digital plane

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 Z 2 , and the subsequent introduction of the topological digital plane 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 Z 2 into subsets of K 2 . This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in Z 2 correspond to topological properties in K 2 . Moreover, we address whether the topological Jordan curve theorem for K 2 can be derived from the graph-theoretical version on Z 2 . Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.