Walk-set induced connectedness in digital spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane for the study of digital images.