Digital Jordan curves---a graph-theoretical approach to a topological theorem
A (graph-theoretical) Jordan curve in a graph is an induced cycle of length greater than 3. A graph is locally Hamiltonian if the neighborhood of every vertex induces a Hamiltonian subgraph. The main theorem of the paper is on locally Hamiltonian, connected, planar graphs \(G\), stating that \(G-C\) contains exactly two connected components for every Jordan curve \(C\) in \(G\). By applying this result to certain infinite planar graphs, as the hexagonal net, or the graph obtained by adding exactly one diagonal in each unit square of the rectangular integer lattice, some digital Jordan curve theorems from \textit{E. Khalimsky}, \textit{R. Kopperman} and \textit{P. R. Meyer} [Topology Appl. 36, No. 1, 1-17 (1990; Zbl 0709.54017)] and Klette and Voss follow immediately.
scientific article; zbMATH DE number 147516
scientific article