Convenient adjacencies for structuring the digital plane

Abstract

We study graphs with the vertex set ?2$\mathbb Z^{2}$ which are subgraphs of the 8-adjacency graph and have the property that certain natural cycles in these graphs are Jordan curves, i.e., separate ?2$\mathbb Z^{2}$ into exactly two connected components. Of these graphs, we determine the minimal ones and study their quotient graphs. The results obtained are used to prove digital analogues of the Jordan curve theorem for several graphs on ?2$\mathbb Z^{2}$. Thus, these graphs are shown to provide background structures on the digital plane ?2$\mathbb Z^{2}$ convenient for studying digital images.